OP, as an LDS convert considering BYU Law, I have to say, it seems like it's a tough option for you. I've had family go to BYU for a variety of degrees across the last 20 years, and have their advice handy:
The Honor Code is enforced. This is not some piece of paper, the majority of LDS members firmly and devoutly believe these things. There is no Bar review of drinking. Pre-marital co-habitation is playing with dynamite. I would not recommend attending unless you intend on complying with the honor code.
In regards to the connections, it will help. In the Mountain West, where you want to work, you will see your degree viewed very favorably. Wherever you move, there will be a strong network of pro-BYU lawyers. Not all will be Pro-You, though, if you are not a member. Some, but not all.
As far as community, the Law School is pretty open and nice. I've heard the first year is very hard, and that you will have some serious gunners, but that was from a grad 10-ish years ago. However, when you can't drink, smoke, or carouse, studying is a whole lot easier
You will have interactions with Missionaries. You will have neighbors who are nice, neighbors who are nosy, and neighbors who have no interest in you.
The costs are staggeringly low, yes. Cost of Living is quite affordable too. Just visit (for like a week or two, at least) to make sure the money you're saving is worth some of the tradeoff you'll be making.