RH, I'm with you in spirit.
My husband and I have been separated with a formal agreement in place for almost 18 months. I caught him cheating and then lying that it was over so i told him to leave. I found out about his CSA about 7 months ago. It was that point when EVERYTHING made sense. I now think this was a "death by cop" situation. We don't have a child so it's easier in ways but it also cut out contact completely.
He finally appears to be really dealing with his past and has been in touch more in the past month or so since I told him I've given up and want a divorce so I can move on with my life. Certain conditions appear to have meant the real and absolute rock bottom for him.
The divorce is on hold while he gets to a healthy place, and i could go either way (divorce or reconciliation) at this point. So really i don't know what the future holds.
My advice to you is to protect yourself and your child financially through a separation agreement and focus on being the best you that you can be, for yourself and your child. Nothing you say or do will expedite his recovery...it can only happen when and as he is ready. As for the lying...I don't know if it will stop or not. At this point I think my husband still lies and gets his back up when challenged on it. I think it's a mechanism for isolating himself and thereby feeling safe. But again who knows what the future holds.
All the best, and feel free to PM me if you need to.