Maybe a way to get the MIL to let go of the Brazilian restaurant is to say to her something like, 'It's Spooky's birthday, so I think we should go to one of her favorite restaurants. Let's save the Brazilian restaurant for your birthday. It's not Spooky's thing." The MIL clearly doesn't get the vegetarianism angle, so maybe the giving-the-birthday-person-a-treat angle will sink in.
In general whether to take a vegetarian to a churrascaria probably depends on both the vegetarian AND the restaurant. In this thread some people know of churrascarias with magnificent salad bars, but some people have been to some with really deficient salad bars. Some vegetarians (such as me, sigh, and I wish it wasn't this way) are really grossed out by the sights and smells of a really meat-crazed environment, and others are not and might even like the smells.
As a vegetarian, I'd put the specific etiquette as "be careful about inviting vegetarians to a churrascaria or steakhouse." It could be fine, but it could be terrible. It's the same for steakhouses. A lot of people think that actually steakhouses are the best places to take vegetarians, because they think all steakhouses have great side dishes and salads, but actually I have been taken to steakhouses where I sat there nibbling on a dinner roll with nothing to order. They don't all have salad bars.