I mean, it's not out of character for him to be religiously infatuated with you, not really, I wouldn't say so.
One of Batter's lines that stuck with me is "Don't do that. I need you to purify this world", or something along the lines of that in response to the Judge's proposal for you to just ditch him.
(I perfer to read the final choice scene in the game as both Judge and Batter being desperate to get you on their side, because the side whichever you're not on is 99% doomed)
And like, it really makes me think just HOW MUCH the Batter relies on you to fulfill his purpose. He can't purify the world without you, can't fulfill his purpose, can't do something his entire BEING is made for, without you. His existence basically has no meaning without you, as in that scenario he is pathetic and useless.
This also kind of overlaps with the Christian idea that a person is NO ONE without God, and can only do the things they do and live their life the way they do because God allowed them to and without him they're nothing, actually.
All in all, 'canon' Bater x Player is not much a ship as it is just literal religious worship. Point is, if you want a "traditional" sort of ship with the Batter, mischaracterization is required, HOWEVER if you're fine with being the centre of that man's existence in the same way God is the centre of a religious' worshipper's existence, he'd be pretty in-character.