Here's what I've realized it's important to remember about LYSs. For the most part, they're owned by people who love to knit, maybe have taught knitting prior to opening their shop, and enjoy owning their own small business. Sometimes it's just one owner, sometimes it's a couple of people who went in on it together. They may or may not have some retail or business experience, but often not.
Which is to say that these are usually folks who never spent time in customer service classes required by a large corporation. So some of them are wonderful, some are not. Some will think, "I'm at least getting the money for another body in the class and if they're happy they'll tell their friends" and some will think, "If this chick isn't buying the yarn as well, she can go take a hike." Some will state policies clearly ("I'm sorry, but every use of the swift takes a little off of its life, you can imagine how many people buy yarn on-line these days and ask for this favor, so we don't wind yarn that wasn't purchased here.") and some don't (the awkward conversation laikabear witnessed).
What I would NEVER do, is just walk in with the knitpicks yarn and keep your fingers crossed. I would call her in advance and say that the price of the yarn is fairly steep for your budget, you'd love to take the class, but want to bring your own yarn, what does she think? I might also offer to lie about it to the other class members-- tell her that if she doesn't want you to tell other people where you got it, that you'll take off the labels and say that someone gave it to you and they said they forgot where they got it from.
And if she says, sorry, you have to use our yarn, I would very politely and pleasantly say, "I'm so sorry, I won't be able to take any of your wonderful classes here, I'm so disappointed." She make re-think it, then!