You could try rubbing pipe dope into all the threads, then also wrapping the new drain piece threads with teflon tape.
The cuts might make the new drain cover hard to screw in.
Maybe buy two, and try to screw in the first one with just a little teflon dope on its threads as a lubricant.
It might act as a kind of thread repair tool.
Then remove it, and install a nice fresh one.