After doing a little detective work on my bathtub handle, I discovered that underneath the plastic cover is a screw. When I unscrewed it, a little piece of plastic fell out and THAT was what broke off!!!!
I am pretty confident that I can glue that piece back on and then re-attach the handle! I just need to figure out what kind of glue will work best.
I am so happy that it's going to be a simple fix costing only the price of the glue and my time.