No reason not to, if it works best for you and is still true to the music.
Yeah it's fine, but my suggestion would be to consult your teacher and find a way for you to play the crosshands comfortably. I mean assume you would like to play Bach and other baroque composers in the future [you seem like the type of guy], and there's a lot of hand gymnastics in there, so it would be better for you start working on them now than later.