The Isis I was playing isn't even the most famous band by that name (a metal-adjacent band from the aughts). It's a Chicago/Blood Sweat & Tears-esque all-female horn rock band from the '70s. Fun but not the most interesting. Hell of a cover though. This is one of the better songs from it.
Ironically, it has a singer who does have a pretty clear Janis Joplin influence. I can see the connection you're making to Alanis Morissette, although it seems any influence is more likely to be indirect - Morissette seems to have mostly cited Tori Amos and Sinead O'Connor as her influences, and those two seem less similar to Joplin than Morissette does to me. (Maybe Pat Benatar would count though?)