@beingtakenforafool
If he's willing to sign over the tenancy get him to do it NOW, regardless of whether he leaves afterwards or not. You must strike whilst the iron is hot. Tell him you won't 'force him out' (fingers crossed behind your back) but that him signing will help you to 'accept things'.
Please, for your own peace of mind, quit talking to him about the 'whys' of the split. He doesn't care, you can't change his mind. All you're doing is poking at your own pain. He has none to poke at.
If you are remaining quiet on urging him to leave because of the DC, that's your decision. But at this point I'd treat him like an unwanted 'ghost' (no typo, that's ghost not guest). For you, he no longer exists in the house. Do not speak to him. Do not cook, clean, do laundry, or perform any domestic services for him. Freeze him out. He's already frozen you out, hasn't he?
As far as being a 'good dad', good dads don't cheat and that's a fact. There is no such thing as a 'cheating good dad'. If he was unhappy the right thing to have done was to sit you down months ago and told you he was unhappy. Instead he chose to keep you in the dark, take up with an OW, and then dump all that on you. Even if his decision had been irrevocable 6 months ago, there could have been counseling to help him tell you in a kinder way and to help you both 'uncouple' in a better way than is happening now.
Also, you cannot rely on his past behaviour to predict his future behaviour. When it comes to separation and divorce all bets are off. He could turn into a greedy and demanding shit tomorrow. See that solicitor, guard your words.