To the point of your post, as you have asked, you probably won't be able to R with your SIL. She obviously doesn't want anything to do with you, she is fresh out of d-day, her H is probably lying a bit, you were drunk enough to not remember things when you told her about the A, and, and this is a big one, her whole family knew and didn't tell her.
You are the last person she will want to hear from. There will like be repercussions for the rest of the family as she continues to process these things.
Your H should be able to talk to her, but he betrayed her too by not informing her of the A.
R is between you and your H, not the rest of the family. If you and your H are R, then you present a united front. In this case, you and your H talk about your own boundaries and your own goals as a family. You keep to that and limit your exposure to people who don't support you. You keep friends of your M and you cut out people who aren't friends of the M.
In your case though, family is intertwined quite deeply. As others have stated, it doesn't matter what others think, but you still have the family to deal with. Quite honestly, they are going to side with your SIL. They all betrayed her, she is reeling, and there is a bit of crazy going on which is completely normal for a freshly minted BS. They are going to support her, not you. This is where the difficulty comes in with your H. He is direct family too. But since he betrayed her by not telling her 6 years ago, he is likely going to have to go along with some of his sisters rules.
To the rules, they may seem very extreme, but part of that is because you've been operating "normally" for six years, lying to her for six years, deceiving her every day of her life by creating the web of false reality. Her reaction is what it is. Are they extreme? Sure, a bit. Are they normal? Yes.
I would have your H talk with your SIL about things like the games and practices. I don't think you should have to move leagues (just my opinion). I also don't think you should drop your child off at family function. This is where the united front idea comes in, and IMO, the united front is that you and your H and kids stop attending family functions for a while. As the SIL processes and begins to heal, you will probably be able to come back into the family fold, but you need to be prepared to wait for a couple years.
Anyway, don't talk to your SIL. She wants nothing to do with you right now.
And, we do get it. We have been where you are right now, with some differences...so all of the "non-to-the-point" stuff which has been brought up is very relevant to you and your situation even though it isn't what you are specifically asking for. We tend to read between the lines a lot here, and for the most part, when you have a bunch of people telling you the same thing, you might need to consider that they know what they are talking about.