He is three and there are going to be power struggles. It is fine except when it comes to struggles that are necessary for you to win, like bathing. I agree with the shower thing, since I could no longer bend over when my daughter was about that age and I was pregnant, I put her in the shower with me and got one of those fun visors from I think Babies R Us, it was a lot easier!
You don't need to bribe him to do something he needs to do though. That let's him win control. Just look at him and be calm and say "you have to be clean, it isn't a choice", "so you can take a bath nicely, or a shower but it HAS to be done", if he chooses neither then you have to just do it. Let him scream, pitch a fit or whatever, reminding him that he CHOSE not to do this nicely. Continue to remind him he has choices about other things, easy things like what socks to wear, clothes maybe, maybe a choice at snacktime however being clean is not a choice.
It isn't being mean but he will test you to the core especially with a new baby around. You have be clear on what is okay and what is not. Caving in will just reenforce his control and believe me you do not want that. Three is a tough age anyway, I never had terrible twos both of mine tested me big when they hit three onto four! :)
Also, once you start changing the baby's diapers, really put into focus baby's pee or poopy in their diaper not big kids, it took a few months of my daughter wanting the attention of her diaper getting changed too but after a call from Santa and empowering her with big girl stuff, then she wanted to be a big girl instead and potty trained 100%.
Hang in there and congrats!