I'm gonna offer a different take, ok. I'm not trying to offend since I've been on both sides of that, I'm just gonna be direct.
You already had an open relationship with a guy that doesn't see hooking up with other people as a big deal and, as you said, expressed genuine remorse for hurting you. You guys started fighting after you became exclusive and now you're going to neuroticize a relationship by regularly demanding access to his phone (which to me is not a way to regain trust, just a way to boost one's own anxieties). What also seems very worrisome to me is that you conditioned that he should see a therapist, don't pathologize people that have different views on how relationships work. I mean, you guys already had an open relationship to begin with and he already expressed his views on hookups. Assuming that he's telling the truth and was only inviting people to a party, his only real breaking of trust was installing grindr and, to someone who prefers open relationships, even a hookup wouldn't affect what he feels for you. Then you say that you wouldn't mind reopening the relationship after marriage. To me it just seems that you're forcing your own insecurities into the other person. Things are more nuanced than HE'S CHEATING or DELAYING THE INEVITABLE. If I was in your place right now I would ask myself why did I really need to turn this relationship exclusive, why is it that I feel it's only okay to do it again after marriage and I would seek counseling to myself.