I have zero knowledge of your fuel system set-up, but doubt you have a non return valve...?
My limited understanding is that there is likely to be a pressure release valve used to not only hold pressure in the system when the engine is initially shut down, but it also prevents over pressurisation of the system when the fuel pump is active, by releasing fuel pressure exceeding the the required operational pressure set point and allowing the excess pressure to drain back to the fuel tank...?
If there is a metal cased mechanical pressure release valve situated within the engine bay, then perhaps it is less efficient when affected by the increase in engine bay temperature, allowing the fuel pressure to drop when the engine is shut down..? .
My initial mail, I ranted on a tad re digital control systems in that on some Porsches and others, operation to unlock the door wakes up the on board systems one of which fires up the fuel pump to ensure that there is pressure in the system to start the engine...
I could be typing rubbish, thus happy to be corrected by anyone who knows the actual fuelling system in this Morgan..
Hope you get it sorted to your satisfaction, but meantime the issue is giving your battery, starter motor, and alternator, a good work out.. which may not be a real problem if your Morgan is like many, may be driven irregularly..
