@laura w,
I am confused the wife have move on with a live in boyfriend and you was in the loop as far as his arrangements for child custody with his wife and all he was slow on is filing for a divorce that might upset the arrangements he had work out with his wife and cost dollars.
Unless you was looking for him to offer married to you in the very near future why would a formal divorce to end an already dead marriage be such a big issue with you?
I can understand why he might be reluctant to get the state involved with arrangements, with special note of the child, that seems to be working just fine for them, without a valid reason to do so.