I'm sorry, NLW. Your xh is still choosing his needs (OW) over those of your kids. That alone is the reason he doesn't have a relationship with them. He is completely delusional if he thinks things will improve when the legal matters are settled.
Until he works on himself and truly understands the trauma he caused the kids I doubt it will get better. Your kids need him to take responsibility. They need him to invest in their relationship without motives and without OW being present. He's not capable of doing that.
I understand the feeling of just wanting him to go away. The dust does need to settle a bit before you can go forward.
I have no advice regarding the funeral. It's such an emotionally charged event. What can you do to try and turn things around, especially with regard to your kids attendance? Can you offer a compromise?