But the point is, your DH told her it was fine! Why would she then need to check with you? Lesson learned for your DH, if he doesn't know, he admits that rather than just pretend it's fine so he can stop having to think about it this second - if he'd just said "sister, I don't know, can you check with Hillsideboy?" rather than just say what would quickly end the conversation and stop him having to think about it.
he needs to understand that he has created a drama where it wasn't needed, because he couldn't just admit he didn't know if you were both free or not.
Let SIL take PILs out. Do something for your side separately, but don't effectively tell your in-laws that DH's word is not to be trusted ! Tell him to be honest about saying "I don't know", not just "what will make this person who annoys me get off the phone as quickly as possible, even if saying that will cause my wife /extended family headaches later on"
The fault in this case does sadly lie with your DH, not your SIL., She did ask, your DH lied that he was sure you were available. You are annoyed that your SIL doesn't realise that your DH's word is bollocks and you are in charge of the diary, rather than you being an equal partnership.