I am not (and never was) foursquare BUT I have an extremely fundamentalist conservative Christian background.
IME, most churches, even the hardcore one, do not go Fred Phelps on the littles, especially when they are trying to make money off the parents. So while, yes, that congregation may very well hear anti-marriage equality and anti-"homosexual agenda" and dominionist messages on Sunday, by and large people totally shield their young children from ALL references to sexuality. So they're likely not going to be preaching against gay people to 3 and 4 year olds, they don't want them to know they exist. Will they tell the kids they'll go to hell? I doubt it.
But here's the thing. Part of preschool is also socialization, and there will be a high percentage of kids in that class that likely DO hear their parents talking about stuff in the news, ect. If you have, say, a rainbow sticker or a darwin fish on your car--your family may be ostracized or seen as a "project". I say MAY because some groups are not like that. But it could happen.
I too put my kids in a preschool run by a church, for much of the same reasons you cite. Small classes, great environment, minutes from my doorstep. However, before I put down my money I attended several church services to gauge things, read backissues of the church newsletter, ect. As it turns out, most children at that preschool do NOT go to the church (I don't even think the director ever was a member, not sure about that though). They do not have an evangelism-through-all-things-associated-with-us approach in a formal sense (though of course, as a Christian, you're supposed to be an ambassador at least 24/7, so in a round about way they do I guess).
But because of my background, I NEEDED to get a feel for the church associated with the school. I felt at home and comfortable with that preschool program. But I ended up dropping a mom's group because of the church they met at and some of the things that I observed during the child care portion. I think this is something you have to judge on a case by case basis.
But I would say, if you attend two services and look around a bit and feel uncomfortable, I would not send my child there. For two reasons--because there very well may be some values crossover into the preschool, but also if they are doing things that you don't feel are ethical or represent values you believe in, why would you help fund that?
It could work, and I would not automatically say no. I think even if you hate church it's one of those things you'll get the most info from if you actually go there once or twice, and see if it's a welcoming, gentle, respectful place. And you can ask if the preschool is considered a ministry or is it run separately.
Anyway, that's what worked for me. I was so close to not enrolling my kids because I was extremely damaged and gunshy at the time. But I'm glad I went to the service, talked to the pastor, talked to the director, and I felt comfortable there. It ended up being a great experience, I have no regrets, even though I have some doctrinal disagreements with that particular church they are good, sweet, respectful folks.