Would it be odd for my bathroom door to open out?

Not at all. I'm in the U.K., and it's considered a benefit if a
bathroom or lavatory door opens outwards when the users are of an "at
risk" group, i.e. elderly, infirm, etc.. If the occupant collapses and
falls unconcious in a small room, the door is easier to open outwards.
I've changed quite a few for people who are living alone, with only a
telephone emergency link.

Besides, it's your nickel (as I believe Americans say).