Insist on NC apart from essential work.it is not up to the two of them to decide. What she thinks, feels etc is irrelevant. If he wants to rebuild the M, he has to agree to NC and then he tells her NC. No ifs no buts, that is how it is. You and he need to stand together. OW has nothing to do with it. She has to be told clearly and firmly its over.
My FWH OW is a coworker and I hate it. Since commencement of R, he has stayed with same company but moved offices, some does not see her every day anymore. Fine has to work in the old office, he has to tell me in advance and if he is expecting to see her for work reasons or not. He tells me when he gets in if she is there or not as she has external meetings and he no longer knows her schedule. He tells me after every meeting with her and I demand to know exactly what he is feeling about seeing her etc.
He also has to travel with her and others which is absolutely awful, only rarely, but is dreadful. This last time, he kept in touch with me all the time, didn't drink, left as soon as the meal was over, sat other end of tale, did not discuss anything personal and avoided all eye contact. He only spoke to her about the work and made it quite clear that he did not talk to her about anything else. I have to recognise he has to work away occasionally, but I make him define his boundaries and report back on them. It's hard but it is getting easier.
I drafted a contract with him which I made him sign. It has worked for us and he has kept to the letter and more, whilst I have not kept to my side. I have screamed and shouted and got hysterical and he has stuck right in there to make me feel better.
Your BH needs to show commitment and do whatever you want him to do to make you heal. In my view, there has to be NC except for essential work and complete transparency about an work contact. He should also aim to cease work contact over a reasonable period oftime