I went through kind of the same thing. My family is a lot more reserved and doesn't drink, while my fiance's family definitely does. We've chosen to get married at a plantation in town instead of a church, and his parents are paying for the alcohol since they are the ones who want it. I know we will have people there who don't agree with drinking, but (as snobby as this sounds) it's my fiance's and my day. My parents would have never paid for alcohol, but since his are, it's all good.
As for a cash bar, I was always told that it's difficult to have one because you have to announce it ahead of time so people have money with them for it, and it's hard to announce it tactfully. But if you want to do that, I know that people do it a lot.
Hope this helps
Good luck!