Not confused at all - with port injection, 50-60 psi is not going to be blocked by gases in the inlet port which will be at or near atmospheric pressure, or less once the inlet valve opens and the engine starts sucking in air via the air inlet, filter and inlet port. Talk of vapours blocking entry of fuel, if as you say originated from the dealer, just sounds like the usual customer service BS when they can't be bothered investigating a fault.
Like others, my experience has always been that engines start more readily when warm. Any effects of a weak battery would be more apparent in cold starting. If that was not the case for the OP, then I still think a temperature related fault is the most likely answer.
Certainly the batt has fully cured 2 x 1250’s that I know of personally. The vapour lock in fuel lines is a known entity and it happens more than you might imagine in bikes. Try starting a hot 4 stroke Mx bike compared to cold, they are a bitch. You generally overcome it on road bikes because you can crank past it. Less so if your batt isn’t up to it.