So MANY good opinions/points here on this thread.
I would take the class using rental gear and continue to educate yourself a little more using this board and other options like your LDS and then add your class/diving experience together with what you will learn. Then, maybe, your combined experience will lead to a more positive gear buying experience. After all, it's hard to know what you really like before you do it.
As for BP and Wings, I will have to admit that if I would have bought a BPW off the bat, I would have saved myself over $1000 in waisted BC's not to mention my time. I love my BPW, and while no gear is perfect in all situations, I have found that my BPW is as close as it gets. Love it, love it, love it! I have used it with my dry suit and with a 2mil skin and loved it! Let me add that MY BPW isn't really all that much dif. than a jacket style BC as I don't have continuos webbing (DR transplate) and only use a small 30# single wing (venture) on my SS BP. One big dif. is that I don't have a ton of material clogging up the front of my body like I used to with my old BC. Still have a power inflator and straps though. Once it's set up, not much dif.
Many will argue that a BPandWing is for tech diving and is too advanced. That is an argument you may hear from a LDS that doesn't sell them. By design, it is simple and that fact doesn't complicate its use.
As for a crotch strap, I like it. Hey, it only costs $5-$10 bucks for a crotch strap anyway so get it and try it. If you don't like it, clean it and send it to me!
My biggest advice to you, once you do decide to get gear, is to buy what you WANT the first time or you will NEVER be happy and end up buying it later. This will cost more initially, but will save you time and cash in the long run. Again though, how can you know what you want if you have never been diving. Once you take the class, it will all make more sense to you. Have fun and welcome to this awesome sport!