Honestly, TTF, your best bet is to answer some of the questions you've been asked here about what you did, why you're avoiding discussing it, and what he's demanding of you.
From the limited information you've provided, it sounds like you and your BH have each other trapped in a standoff where neither of you can start healing. You're sticking to an implausible story and white knuckling it through periodic abusive behavior, motivated by shame and the hope that he'll eventually wear himself out. He's casting around for toxic solutions (force you "to submit," beat up AP, fuck someone else) that will not solve anything and indeed have great potential to pour gasoline on the flames. The good times make you believe that you've turned a corner and are putting your infidelity in the past. He senses this and is infuriated by it, and the cycle begins again.
I get it. When all of your lizard brain instincts are signalling fight or flight, it's agonizingly difficult to be deliberately vulnerable. If there weren't still some love between you, you wouldn't be so afraid of the power you hold over each other. You're both scared of hurting and scared of being hurt, and the resulting defensiveness keeps driving you further apart.
As counterintuitive as it may sound when you're trying to hold tight to your marriage, a short separation (one in which you do not see other people) might be a good idea here. You need space to get serious about confronting your suppressed memories and coming fully clean. He needs to find healthier coping mechanisms. The roller coaster is a natural response to infidelity, but for the sake of your kids, if nothing else, it sounds like time to get some distance and perspective.
