Remember that the ECU internals do not run at 12V, they will be most likely 3V3 supplied by a controlled DC-DC, hence if the volts available is enough to crank, its more than enough for the ECU to run without any issues 
Yes... But it's seeing 11V and having a hissy fit because of it. connect jump leads and it starts fine. Never understood why but seen it before. Last one was a ground fault. Previous was starter internals.
Sorry, disagree here.
The original symptoms state:
I've worked out the issue is a "hot start" problem. When you start the car from cold, it almost catches, then doesn't. but the next time you turn the key, it's off - no repeated cranking.
and
The car will then run all day long without missing a beat. But when you turn it off, having got up to temperature, it WILL NOT start - it just cranks over and over
Which, given that the worst case for battery terminal volts is after a period being stood and cold start, goes totally against the diagnosis some are stating. This is further backed up by the fact that even if there was a very large initial voltage dip whilst the starter engaged, the ECU would be up and running within 1-2 seconds after this assuming a watchdog event on the processor had occurred (it wont, there is more than enough smoothing caps and holdup in these units to ride through a few milliseconds of extraordinary voltage dip).
Remember all, test and diag is the key to effective diagnosis, leave the willy nilly part changing to the professionals lol

So questions have to be
1) Is it turning over ok (removes the battery from the equation)
2) If you continue to crank does it start (further removes the battery from the equation)
My money is still on an electronic component which is struggling when hot, classic symptoms of crank sensor actually and giving a few more volts is resulting in a few more rpm and just getting over the threshold of speed where it works. Sadly this is also the symptoms of a goosed Siemens ECU! Change the battery and you might mask the issue for a few months until the morning temperatures drop again.

So much more diag needed