I think before binders became more popular, people used to use bandages. Like medical bandages and just bind those around their chest. You probably need to get a certain amount of knowledge and practice for that to work well, but it could be a low budget way to do it, if you reuse the wrapping. Or maybe something like sports tape/wrapping could also work?

It's also more inconspicuous, because if someone asks why there's medical bandage stuff in your room, you could just say that you're looking into first aid and trying to learn a bit about first aid methods, because it's just a good practical skill to have.

And as for change in looks. Sweaters and baggy clothes can hide your body shape and they might not even notice if you're a size smaller suddenly while wearing that.

If the changes are continuous and subtle, they won't notice the accumulated change, because they only see small increments.