So you have a tricky SIL (boat rocker) who your PIL are scared off and so they use emotional blackmail to get you to be a boat steadier along with them.
The only way to deal with a boat rocker is to acknowledge that it is not you rocking the boat, but SIL, and you are not going to enable her behaviour. Say NO, and say it lots. I have a similar situation and I have learnt to say things like "that doesn't work for us, sorry" and "this is what WE (me, DH and DC) are doing". Once you start saying no, they lose all power over you.
That aside, in a tricky situation like this, I would let the RIGHT THING to do take the lead and the least fall out.
So, it sounds like your in-laws don't really like you that much TBH. So, whether you go to the wedding or not, you are still going to be the unappreciated DIL. What is worse for you, losing your friend, or your in-laws being upset? Which is worse?
You have committed to your friend's wedding. You should go. As a compromise, your DH should go to his sister's wedding.
Your SIL should have asked if everyone was free. 6 weeks is no lead time whatsoever. People could have been easily booked out. Is it a small low key wedding? If so, she should have asked about the date, and if people couldn't make it, she could have changed it to suit everyone.
Your SIL sounds like a bit of a nightmare, and if you don't stand your ground with her and your PIL, then this will just be rinse and repeat for as long as you are all around.