Please do not confuse what “The Way” does, with being the general practice of the Catholic Church. I don’t want to bash on The Way because maybe it is a good thing for some people, but they have been criticized for, among other things, objecting to relationships their members had with people not members of The Way, even if the person who wasn’t in The Way was a practicing Catholic rather than a non-Catholic Christian. The Way kind of wants to do its own little stuff with its own members from what I’ve heard. It is not an inclusive group.
I’m not sure if them holding a “private Mass” where they decide who can attend is allowed. In recent years, the Church has frowned upon excluding anyone from a Mass, which usually happens in the context of groups wanting to have some “private Mass” for their members only. You could ask the parish pastor whether it is permissible for them to exclude you in this manner, although I realize maybe you don’t want to make waves with your boyfriend’s fellow Way members.
Regarding regular ordinary Masses not held by The Way, you’d be welcome to attend them, although since you are not Catholic, you’re asked not to receive the Eucharist. My husband is not Catholic and has been to many Masses with me, no problem, once they even asked him to help out with taking up the collection.
I hope this helps. God bless you.