I think it makes more sense for her clothes to not transform at all. I understand your desire for modesty, but it doesn't make sense to only have some of her clothes transform, but not the others. You definitely need to address it, but it shouldn't be a central part of the entire story.
You could have a humorous scene when she first transforms and loses her clothes. It can be funny and embarrassing, but not scandalous. Then, once she realizes whats happening, she can plan for that. Think about the old Spider-Man comics where Peter Parker would stash his street clothes in an alley or a garbage can and then go back to get them after he was done crimefighting. Brianna could do something like that.
After a few issues, that could just be the standard way she handles it, your readers will know it, and you don't have to address it every time she changes.
I think a key question is how well can she control her transformation back to human. If she can, then she can stash clothing or just know where every thrift store with a changing room is located in town.
Either way Kevin is right, once you and your readers know the rules then you can have some fun with plot twists.