I went to Regent's Park.
I'd say you should go there. To be honest, it's not ideal, but it's your best option. It was the same for me. I applied to what some people would call "a proper college" but was offered Regent's so I took it. I'm currently stranded in London and yearning to go back to Oxford to do a DPhil (I already have a University of London MA with distinction, but I'm told that Oxford will consider that roughly equivalent to an Oxford BA!) and I would be hoping to do that at somewhere other than Regent's - Christ Church and Merton are favourites.
What you'll find at Regent's is quite a lot of public school people (the posher the school, the more riotous the behaviour, Winchester and Eton being the worst/best depending on your perspective) who couldn't care less about religion, get drunk, smash things up, and sleep in the quad (the one thing the principal doesn't like - running round to quad naked is fine, as long as it's done quietly). Then there will be the more nicely mannered public school people (often girls, or boys from lesser public schools), who are ardent evangelical Christians and drink moderately. The college Christian Union is quite big and active and its outlook is very conservative - praying for the college fellows because their interpretation of Scripture is unfaithful, praying for friends not to convert to the Catholic Church - I've witnessed the first myself and heard the second rumoured to have happened. There are some people who are too pentecostal for the CU (speaking in tongues, prophecies, etc) - they tend to go to Oxford Community Church, they have cell groups, and they usually don't drink.
However, you'll find that a good 50% or more are just ordinary students, and if you take a look at the weird behaviour that goes on at some other places (sex in lavatories during guest night dinner at ChCh was reported in one of the student newspapers) you'll be pleased to have the sober influence of a Baptist ethos. Actually, I should point out that among the regular students (not the part time ones who visit to do Baptist studies) there are very few Baptists - about one per year at most in my time.
The last thing I'd say is that if I had to choose by list of ideal colleges Regent's Park would be somewhere in the middle. Merton would be at the top, followed by ChCh, Balliol, Magdalen, New Coll., Trinity, Univ, Exeter, Lincoln, Jesus, BNC, Corpus, Hertford, Worcester, Teddy Hall, St John's Oriel, Wadham, Queen's, Keble, Mansfield, and St Peter's, but I'd choose Regent's Park over LMH, St Anne's, St Catz, St Hilda's (now that it's taking men), St Hugh's, Somerville, and all the other PPHs.
Greyfriars and St Benet's have the best food I ever had in Oxford. Indeed, St Benet's would be my favourite college if it had women.