Car won't stay running

fuel pressure isn’t needed to charge the battery, when you say wont stay running is the engine stalling whilst running or are you having problem starting it each time over a period of a week or so?

if its stalling and won’t hold charge then get a multimeter and check the battery whilst running, if the alternator is running fine it should read out between 13.5-14.5v, if its lower than that then that may be an issue. if the battery isn’t holding charge and has gone below 10v numerous times then the battery will be damaged and needs replacing.