I think the problem is the mat you have is too flexible. The mat I have by my sewing machine isn't particularly flexible compared to what you have. It stays put.
For safety's sake, I would take the mat you have out of the sewing room. After all, if you wreck the rug with the chair, you can always get a new rug. If you trip and seriously hurt yourself because of the mat, well, it's impossible to get a new you!