Congratulations on working through it to success.
Reminds me of the challenges I went through for an apparent "over- pressure" problem on a HIF44 - two different ones actually.

First photo is the collection of components to go from the mechanical pump to the carb, including a regulator and in-line filter. The black line is the cross-over pipe covered in reflective insulation and cable wrap to hold it in place - an attempt to eliminate the possibility of vapour lock or hot fuel. The long pipe at the bottom is the crossover pipe routed below the bulkhead cross-member, used when the mechanical pump was removed and fuel temp/vapour lock was confirmed to not be an issue. 

Second photo is of the installed setup. What it doesn't show is the change from the mechanical pump to an big Facet electric one in the boot, to eventually a proper SU pump under the boot where it should be. (If the Facet sucked air, it sounded like a machine gun!)

Third photo is of the current and successful setup - no regulator and just the filter in the engine bay where it is easy to check condition and flow and eventually replace.

My real problem: Float needle valves (both carbs)  with insufficient tolerance - when they warmed up, the needle would expand just enough to stick closed or open.

Most people are probably not aware that the fuel level in a Mini tank when nearly full is enough to prime the system right up to the carb bowl. I found that out the hard way, when the float valve stuck open and fuel drained right through the carb and down into the engine sump, while the car was put up for the winter. Now I never store it with a full tank. 

Success yesterday and must say that the 1293 is now singing nicely - even at WOT.  Took him for his first shakedown on public roads - which was scary enough and quickly produced a scraping noise due to a slightly bent rotor dust shield at the front (and quickly bent into shape).

So we started looking at the fuel pressure with just the small generic inline filter and kink at back of tank and got all sorts of wonky readings....started out a 3psi then went to 7 and back down...so we quickly fixed the kink and removed the generic filter.  Went back to the Filter King and now started to hunt for the right psi.  It was easy at this point to lock it down around 2.5psi, but required to adjust that regulator a lot more than what we did a couple weeks ago.  You get ZERO help with the FK instructions and clearly we should have started with the pressure gauge from day 1.  Lesson learned.

Once we got that locked in, fixed at the same time the left SU as it was overflowing and leaking...a few manly bangs on the side of the float chamber, settled the needle back into shape and bob's you uncle.  Ran him outside and next to timing and vac.  All checked here...had some weirdness with the Swiiftune CSI ignition - could always get 10 degrees at idle, but hard to get right timing at WOT - so did a quick reset and picked the right ignition map and done there.

Pressure is steady and runs so much better now.  Pictures attached recapping the day of "find the right pressure" games.  Even had some time to get a badge and bonnet strap into place.  Really chuffed with how he is running....still have a slight fuel smell in boot even with vent line secured with a nice sized (and tight) grommet...not sure if this is just par for the course...thanks again to everyone's help.

Edit: Decided not to go with the electronic pump....staying with the mechanical for now.  Works perfectly.  Also need to invest in a fire shield (fuel line protector) where it is now on back of engine...just in a very hot spot and want to protect that line from the extractors.

The mechanical fuel pump does not need a regulator. Correct the kink in that line, check that the tank vent is routed neatly and open, and I am sure the engine will get fuel as it should.

Thanks Doug - appreciate your thoughts.  May hit you up later

LOL - yes.  Taking care of that soon....I will start with that and then work my way forward.   Will see how he behaves without the Filter King first and may just remove that and use a standard plastic inline filter.  Not sure how much a regulator is needed is I stay with mechanical pump....but with electric pump, may need to regulate back a bit.

Is there still a sharp bend in the flex line coming off of the tank? 

To address one of your first questions, you don't need to do anything about the pump's GPH value.  They specify the MAX flow the pump can deliver.  That isn't what the pump flows all the time.  Every time the float bowl gets "full", the flow basically stops.  The fuel will only be delivered at the rate the engine can use it.

It sounds like your plumbing scheme is OK.  As with electricity, start with where the load is and work your way backwards to see where you are loosing flow.  In this case, disconnect the line at the carbs and look for fuel flow.  If you have flow there... the problem is inside the carbs.  If you don't have flow, move backwards from the front to the back checking for fuel flow at each place there is a connection.

Sure - for my Mini it is:

Tank (original) ---> Rubber Fuel Line  ---> Metal Fuel Line ---> Mechanical Pump ---> Filter King ---> copper carb line split to HS2s.
[Adjusted fuel pressure on FK blindly, yes, not the right way to do this]

to now:
Tank (original) ---> Rubber Fuel Line  ---> Metal Fuel Line ---> Mechanical Pump ---> plastic filter ---> copper carb line split to HS2s.
[Barely filled the small inline filter and had to prime]

To tomorrow:
Tank (original) ---> Rubber Fuel Line  ---> Electronic Pump (Delphi in-line unit) ---> Metal Fuel Line ---> Filter King (setting at 2-3psi) ---> copper carb line split to HS2s. (Fuel Pressure Gauge to monitor pressure after Fuel King)

That's what I am thinking anyway.

Can you clarify your' order of operations....?

 Tank  --- >   Filter ----> Mechanical pump ---->  Carb???

If thats the case i have my own doubts the mechanical pump will draw enough THRU the filter assy?


I Have a friend with same filter assembly.

 He runs =  Tank   ----->  Facet Square Low Pressure Pump ---->>>  Filter assembly = Carbs.. Works flawlessly.   you're using the filter + regulator to ' regulate the pressurized fuel from the pump.

Yep - just did that....primed new lines and finally got him to run well....but after sitting for 5 days, was hard to start and run...so had to re-prime again....seems like a electronic pump may help solve for pressure and times for when it sits.  Just my hypothesis.

We have had the line within the fuel tank clog from the car setting. Also, on a Cooper S, we have had to clean the crossover line, and clean the line within the tank too! Easy to diagnose when you disconnect the fuel line and nothing comes out. I now have a special tool, made from heavy wire to clean the lines.

 

My stock A+ mechanical pump was taking a life to prime after new fuel lines. I had to draw the fuel through the lines, pump and filter with a little vacuum, then all was fine. I have no stand alone regulator on mine.

Getting some weirdness from my mechanical pump or from the fuel filter/regulator (Filter King FPR005) I just installed on a twin HS2 1275 setup.  Just not getting fuel into the filter and not sure if it is pressure or a gph/volume issue.

Tomorrow me and my buddy will get a fuel pressure gauge on the lines to see what is happening.

In the meantime, I picked up this Delphi electrical pump from Napa as a backup measure...I can get the psi down to 3 or so with the regulator, but is the gph too high here?  I assume there is no way to reduce the gph.  Just wanted to make sure before I connect this to see if this gives me stable fuel delivery with the Filter King (vs mechanical which may be faulty).  And yes, I have a rubber fuel line kink that I am correcting (from another post last week).

https://www.napaonline.com/en/p/DFPFD0037

Average Current Draw	2 A
Average Flow Rating	27 gph
Fuel Type	Gas
Gasket Or Seal Included	No
Inlet Fitting Thread Size	1/8 in. - 27 NPSF
Inlet Type	Female Threaded
Maximum Pressure	3 psi
New or Remanufactured	New
Outlet Fitting Thread Size	1/8 in. - 27
Outlet Type	Female Threaded
Pump Location	External
Pump Type	Electric
Strainer Included	No
System Pressure	4 psi