Bud at that voltage its pretty clear the battery is healthy.
I had exactly this issue when i had my 335i turned out to be HPFP.
The story of LPFP running at low RPM when starting is BS, they both work together, although the lpfp is pumping, the HPFP is the one that does majority of the work, including cold starts, this does not mean to say your issue is the same could be LPFP, my advice, get your self some wynns fuel system cleaner and run that with a tank before throwing money at the car, extremely rare the regulator goes, good luck bud.