At 2 gallons in 12 minutes, that is only 10 gallons per hour flow rate, which is way too low. You are indeed running out of fuel at speed as not enough is being supplied. Time for a new fuel pump. A Mr. Gasket 42S electric fuel pump works well - 2 to 3.5 psi and 42 gallons per hour - and is readily available through any of the major local parts shops (O'Reilly, Advanced, Autozone) for about $50.

All fuel lines cleared and changed.  Pumped 2 gallons of gas into a can in 12 minutes. 
I had a mechanical advance dizzy originally with the car.  Changed to petronix vac advance dizzy & new coil new cap and wires- exact same issue remained. 
I have done the timing.  tested it with and without the advance hooked up-  exact same problem persists.  like clockwork.
I've tried multiple needles- the problem doesn't change.

I wonder if I need a stiffer piston on the HIf44 carb? 

when I cruise at 60mph, I'm barley pushing the gas pedal.  I tried mashing down on the pedal briefly yesterday but the problem just got worse until I pushed the clutch in and coasted to let it settle.  Could the piston be in a bad position? (it moves freely)

I'm ready to be done with this issue and don't mind spending some money, but I just don't want to buy another dizzy and coil or anything else if already bought to no avail.

The main part I have not replaced is the carb.  which i would happily do if it'd solve my problem. 
do I have to cut the bulkhead to mount a webber? 

thanks for the input guys.  please keep it coming.

I've read more than a few threads on the net with people dealing with a similar problem.  but none that i've read have a solution posted.  they just fall off the face of the earth.  If i solve this I'll post it.  promise!

to me, electric fuel pump is good for spare and for emergency, in case the manual fuel pump fails, but for everyday pumping? i'd still rely on the good ole manual fuel pump. my mini is 39 years old, and it still got the original manual fuel pump. how's that sound?

fuel system cleaner is the key

I had a pump = Do this == Not keep up above 50 Mph basically... 1275 engine = single 1 3/4 SU. 

     The pump still clacked and still appeared to pump= so I didn't throw it away = but i replaced with a Facet Electric. Done and done. 5 years ago.

   Then that "used" pump then went on my Moke = 998 - Single 1.5 Carb... And it worked.. Less fuel total volume required and ity kept up for 2 years... then finally = Wont run over 50- 550... Now replaced with another Electric Facet Type . Motoring again.

  New Pumps are cheap.... I even carry an extra now in my spares bag...


Pump

Pump
 
Pump

The big question is, WHY does the car run good on and before 55mph and starts to have this "3cylinder running like" phenomenon as the OP said past 55mph? do you mean that the electric fuel pump runs normal at low speed? and starts to fail at high speed?? manual pump is more likely to do so, i really don't think it's a fuel issue. i have a gut feeling it's more on distributor issue., or electrical in general. And pls don't tell me the reason is old gas,,,i have a seadoo boat that seated for 2 years and it still run excellent on old stale gas. I use KN filter as well for the past 1 1/2 years and my mini runs good.

myth

Pressure has nothing to do with much... Too much will cause an overflow past the carb bowl needle jet..otherwise its not really of concern.

Volume is what you want.  Each HP your car produces needs a certain amount of fuel.  Vizzard - and no doubt some other knowledgeable individual will provide the formula.  Your electric fuel pump will provide a constant output if allowed .. Ie if not restricted by the float valve...

To determine if your pump can supply enough fuel, remove the fuel line from the carb, place end in a jar and switch on the ignition (don't energise the starter The pump will run at max capacity.  Run the pump for, say, 30 seconds.  Measure the amount of fuel dispensed.  

If your engine can produce, say, 60 HP, then it will need "X" pints per "Y" seconds (refer Vizard). Compare this with the amount of fuel your pump provides ....

If the amount of fuel provides is less than that required to produce the amount of power needed to drive your car at 55 mph the engine will draw down the fuel in the float bowl.  As the level in the bowl drops the mixture leans out and the engine will start to miss/stutter....

If the fuel flow is reduced (fuel line is pinched/vacuum is forming in the tank because it isn't venting???) the car will run OK at 30mph but will stutter and object if you ask it to do 50.

Cheers, Ian

Paul Teggler's article that you posted the link to was one of the sources I was thinking of when I posted my message above.  As suggested in the link, first try a bit of choke to see if the missing at speed problem goes away.  If that helps, try the stock filter or give some thought to changing needles.  Needle changes with the K&N filter change are not unheard of.  When you buy a Stage 1 kit for a Mini it typically includes the K&N filter and a new carb needle.    

However, the symptoms you describe where the engine stumbles until you give it 5 second at idle speed to catch up suggest fuel supply problems, not necessarily a bad mixture.

Hi guys.

Thanks for the input.  this morning I replaced all rubber fuel lines, and ran chemtool and blew compressed air through to hard lines. 
the filters are clean.  including the one in the hif44.  checked it yesterday. 
I'll rest drive it a little later.

BTW.  This problem happens like clockwork.  very time. 

the car has never run right at high speed since i've owned it.  although it came with non running.  all i did at first was set the timing and install a K&N filter. (the car didn't have an air filter when I got it)


please check out this artifice.  let me know what you think about the filer situation and also the timing advance situation. thanks.

www.teglerizer.com/mgstuff/a_stumble_at_cruise.htm

Also is the carb vented correctly. Some owners or po's have trouble connecting the hoses correctly on HIF's and also a blocked carb vent would create this issue btdt. AS Doug stated you sound like you have a fuel delivery issue so if everything checks out take a look inside the float bowl.
Next time it splutters to a stop just shut it down and remove the fuel line at the carb and check to see the flow, have a container ready.

Above it was stated that there is no such thing as too much air... that's true but also subject to misunderstanding.  There have been a lot of A-series people mentioning over the years that when they changed to a K&N it upset the mixture and that to get normal driving back they had to change carb needles.  In effect, there was too much air for that particular needle.  

Morninglight, when did your car's engine last run right and what have you done to/with the car (in addition to your list above) since it last ran correctly?

Regardless, being able to coast for a short distance followed by normal driving for a short distance suggests a fuel delivery issue.  Check the tank strainer as mentioned above.  Also look for any inline filters installed by previous owners.  Some HIF float valves have a built in/on strainer.  Make sure that is clean.  While you have the carb apart to check the float valve, make sure the float level is set correctly.

(there seems to be an echo in here...)
You say the fuel pump delivers 2-3 psi. But that is probably static pressure. It does not indicate the pressure under full flow. You may have a restriction in the fuel line - tired piece of hose or a pinch in a metal line, or the tank mesh strainer as noted.
Analogy: turn on a garden hose just a bit at the tap. With the nozzle closed, you soon get full pipe pressure. But if you open the end of the hose without opening the tap, flow drops to a trickle. Your car may be getting enough to cruise on, but beyond that, you drain the bowl. When you checked the filter, it would have gas in it even though the bowl ran dry.

Remove and examine the spark plugs. You'll see if youre running rich or lean or etc, spark plug never lies.
A gas tank is designed to have a tiny vent hose situated on TOP of the gas tank that extends all the way to the charcoal canister, it takes a century for it to get clogged , dont localize your troubleshooting on fuel department alone. You said its been happenning for a year now? Check everything, get the compression on all 4 cyl as well.  do you still have the old style dizzy? Replace the condenser too.  Take the distributor cap and examine for burns or anything , it might be arc-ing at high speed. 

 Just test drive with the gas cap off ( or partially covered).  If symptoms disappear, then you have proven it.

No such thing as too much air through the air filter. blocking the filter could cause a choking action, which would ritchen the mixture and may help your problem, but that is not how to cure it permanently. It also sounds exactly like a blocked tank vent to me. You should try installing an accurate (widely calibrated) fuel pressure gauge right before the carb fuel inlet. sometimes fuel hoses will degrade internally and cause intermittent or temperature influenced blockages. Intermitent blockage around the needle/seat (a piece of hose floating around in the seat opening) may cause a repetitive, partial blockage. The list is endless, but is probably fuel feed related. Let us know when you find the culprit.

The strainer in the tank is probably partially blocked use a good fuel system cleaner that will remove varnish build up , seems to have fixed a similar problem in mine.

Letting too much air in? that would be a first for me..................

i just came across an article written by a fella with an MGB with a similar problem.  He said after 2 years he discovered his K&N filter was letting too much air in.  switched to stock filter and fixed the problem.  I'll try that (or block off a portion of my K&N) tomorrow morning. fingers crossed.

IF you have a Vacuum advance, see if it's working or if the tubing is attached, or if the 2 springs in the mechanical advance has fallen off...check them

I've checked the vent.  It's clear but i'll pull the tube off and recheck and retest.  

It is an electronic fuel pump. at the rear..

I have also placed a clear fuel filter right before the carb.  While test driving I cut the key to turn off power to the pump to see if the filter had gas in it. (when it started to sputter) It did have fuel in it.  But i'm still not convinced it's not a fuel delivery problem.  

Is the fuel pump electric or Manual?

If Electric = Is it positioned back near the tank?
   Primarlly electric pumps are Pusher pumps.  Work best pushing fuel to the enigne- FYI

I would also REPLACE the pump if electric = quite simply = Could supply normal volume of fuel OK under slow conditons, but not enough volume to keep up with speed driving.