Does anyone recognize this bathtub faucet part?

hI, first post, looking to be more of a DIY guy.
I needed to remove my bathtub faucet handles, as some salt like gunk was accumulating inside the handle mechanism and really slowing down the flow. Unfortunately when removing one of the handles I broke a plastic piece within the handle, making it inoperable. I'll attach pictures. It's an old faucet set, not sure of the make. Looked in stores and online for part but to no avail. Any help greatly appreciated! Right now I'm using the one good plastic piece to go back and forth from hot to cold.. Good news is the flow is good again.
Thanks!
Ken