In my opinion, "making it about the OW" means placing all your energies in her instead of yourself. What is happening with you and your lawyer and the case does not fall under this umbrella. You have to do what you need to do to make sure you and your children are provided for. Your L sounds like he knows what he is doing!
Yes, it will make your H mad. So? Look at it this way: If you kept her out of it and took less than you deserved/needed because you were trying to keep your H "happy," how are you going to feel if he then goes on to marry her? You're gonna be pissed that you didn't get everything you were entitled to, and it will be too late to do anything about it.
You owe your H no explanations. Just be pleasant and stick to business.
Yes, take someone to court if you need the support. No, not your MIL! Judging by her words while you were gone, she is not a good support system for you.
I know you don't want your H angry with you, but most likely he will be during this time. Just keep working on YOU and your responses (or lack of) to his antics and things will get smoother in time. Right now PROTECT YOURSELF!!!