Lots of context en route so buckle up.
My BF (21M) and I (20F) have been together in some capacity since we were 18 and 17, and have spent three years of university dealing with each other in some way. We initially dated for almost two years after a weird 'situationship' period our first semester of college. When we had to go do our own separate things for the summer, he had begun acting suspiciously about a specific girl in a summer program he was in. This upset me, and I would constantly bring it up in ways that were not necessarily healthy for anyone involved, but he also wouldn't do anything to stop spending time with her. He swore it was friendly, he would never cheat, etc. Eventually, he was so worn down having to defend himself about this that he decided we needed to break up. It is important to note that when we were breaking up, he said that he would not get with anyone until we talked again in the Fall, and especially not the girl I was worried about.
We were no contact for all of August, and then talking and hanging out once a week in September/October. We started hooking up again in September and had a casual relationship that he kept a secret from everyone until late December. By January and February, everyone knew that we were seeing each other. We did not become officially "boyfriend and girlfriend" again until February, after a big fight where he admitted he was worried to commit because he did not want to fight again like we did last time.
Throughout our building back phase, he swore that he had not gotten with anyone while we were broken up, and even said that he was glad we were back together and was hoping for this to happen. Here is where I fuck up majorly!!! Ignorance truly is bliss. After he had a weird bad trip and a breakdown, I got worried he was keeping something from me, and searched the girl from the summer's name in his messages while he was sleeping. I found out he had sent pictures of her to his friends, bragged about how she had a boyfriend and how much he loved when girls cheat with him, and that he was swearing he didn't want to get back together with me at all. He was open to his friends about the fact that if I ever found out, I would freak out.
I later DMed the girl, who told me they had hooked up for about a week and a half (so not just one time) and the last time they had texted was in January.
This is where I'm conflicted. All of the messages I found were sent in August, which was right out of our breakup, and he said that he was very lost and confused during this period and didn't know what to do. He swears that this all means nothing, and showed me that every text he had sent to her (which were few and far between) after August were very professional/related to their shared occupation. He has apologized profusely and is constantly asking me what he can do to right this. He said he lied to protect me, but feels horrible about it and is very understanding of how hurt I am.
I do not really know what to do. The only thing I want is to be able to forgive him and forget that it happened. On the level of technicalities, what he did with her was not fully wrong, and I have set series boundaries around lying. I don't want to be without him, but when I am with him I cannot stop thinking about this. Any advice on how to go forward and forget about this, hopefully with him involved?