I think Frank is quite possibly the true tragic figure in Outlander. Through no fault of his own (other than that he is boring and pedantic, but I don’t think that he can help that), his wife has an unbelievable adventure in the course of which, she meets a man who is much more suited to who she really is than Frank can ever be.
Is he bitter? Yes. Does he have a right to be? Probably. Did he cheat? I believe he did despite DG’s attempts to retcon that part of the story.Is his cheating justified? I don’t know. In today’s world I would say they should get a divorce and try to move on with their lives and find some kind of happiness. But that wasn’t the world of the 1950’s.
So maybe cheating on Claire was his attempt to find someone who wanted him for himself, rather than a wife who he knows is only with him because she has to be.
I don’t particularly like Frank, but I can empathize with him given the circumstances and the mores of the time period.