If you have to introduce a cover song by name and artist because people aren’t going to recognize it, just don’t do that cover song. Don’t waste a “what’s this” moment with your audience on someone else’s obscure crap. Do an original instead.
If the audience isn’t going to recognize it, they’re not going to be impressed, and you’re basically doing someone else’s promotion for them. Promote your own band’s sound by doing your own songs and your own takes on well-known songs.