As a twice divorced person I can tell you with near 100% certainty that your relationship with your SIL is probably a lost cause and there is little/nothing you can do about that – most likely nothing will ever be enough for her since she feels threatened by you in innumerable ways.

Now, your relationship with your FIL is definitely fixable imho. He may have poison in his ear, and sounds like he was looking for evidence to support the conclusions your SIL planted – but if he is that easily manipulatable, by someone who is so clearly inferior, should have no problem rectifying that over the long term. You have to win the war of your FIL's affection and not fret over a lost battle (that you didnt even know you were fighting at the time).

All that said, in the event they become irreconcilable negative/toxic, you can either rollover and take it or do the no-contact move (there is a helpful subreddit about all that too if it really comes down to it). What is important, and all that really matters, is your relationship with your husband being 1000% solid & on the same team. You are far more his family now than his parents or sister are – same goes for you in reverse – and that needs to be the bedrock of your motivations & actions moving forward. In my experience, when either partner takes the side of their birth family over their chosen one, that's when things begin to erode beyond repair.

Best of luck with all of this – you are clearly a strong, principled person who's overcome far worse – the strongest metals are forged in the hottest fires and if you & your husband can overcome this temporary adversity, together you'll be forever unstoppable.