I've read a few of times about failure of a unit. They're likely sturdy units but abandonment to the elements without consistent usage might open the door to their failure. A faulty ground connection is my next choice. A bike starts up, runs then heat and vibration break the connection sooner or later.
I referred to the fuel pump mounting because if a pump slips downward far enough in its mount to hit bottom, fuel flow through the basket style pre-filter can be sufficiently constricted to cause a stall. My experience was with an intermittent stall condition that was eliminated when the pump was raised sufficiently and secured.
I clinically went through all connections during the refurb and inspection of the loom, I re-soldered all that I could. Additionally I ran an extra 1mm earth from the frame earth point to battery and onto gearbox to counter-act the powdercoat.
I found a leak in my tank! :( It was in the right hand side upper part of the ‘step’ seam. It took a long time to migrate through and has blistered the paint badly overnight. I’ve siohoned the fuel, removed the tank, stripped the pump components, ordered a full seal kit and left it to evaporate off the residuals. Grrrrr.
With the MOT this Friday I had to progress the fault finding. I Jerry rigged the pump into a bucket, I made a rest out of some of the spare mesh I had from the radiator mod, this lifted it off the bottom, I then routed it in and added some juice. I pulled the plug from the Ignition Control Unit in the head and attached the old ICU and went for start whilst twisting the connection into different positions during start- this identified the plug as the problem as you suggested, Laitch! Well done! Twisting position dictated fire/no fire.
I heavily dosed both male and female connection to the ICUs with electrical cleaner and vigorously cleaned with a paint brush, left them to dry then repeated the process.
Once dry for a second time I plugged back in as standard position and started. The bike ran with no issues- she heated to 80c and the fan kicked in and out as needed repeatedly. All seemed good. I ran her up through all the gears, no complaints, a happy Brick.

There is some tuning to do with the Acewell, considering the instructions are in English, they are as difficult to make sense of as the old German BEP instructions. The rev counter/speedo is all over the shop when I rev etc etc. Unsure if this is potentially all related to the known BEP fault. I have been informed I need to purchase a new BEP and return the old and apparently they will refund me for the backlight issue.
I have 3 days left for the sealer to arrive. Repair (hopefully touch up the paint) and ride her to the inspection (MOT)
This forum, as usual, saves the day. Click the brick, please.
In regards to a bottomed our fuel pump, Laitch. I’ll look at this. Great idea to lift it slightly! Will take some measurements tomorrow.