I feel like your fiancé wants to believe she’s on agreeable terms with her mom, but if this is happening then that’s simply not the case. You’re looking at hard, empirical evidence that something is still wrong.

This woman is in your home, taking advantage of your hospitality, and then smearing your name when you aren’t looking. That’s far from agreeable.

To me it sounds like your mother in law is somewhat volatile and your wife doesn’t want to set her off. You can ignore it if you want, but if your wife’s prior history with her mother is any indication then this probably isn’t going away.

You don’t need to kick her out, but you DO need to call her out. Preferably with your wife’s support. Maybe sit down together, let her know this is something you feel strongly about, and come up with a way to approach this that you both feel good about.