I concur with others, there's some shenanigans here, the Orthodox boyfriend isn't being fully honest with LW1. I am also detecting more than a whiff of lovebombing here, with all this 'breaking down walls' for someone who has never previously had a relationship.

There is no prohibition on non-Jews attending Jewish holidays, so there's something else going on. The worst case is as librarygeek pointed out, some kind of bigamy situation. A slightly less horrible possibility is that the boyfriend doesn't want anyone to know he's dating LW, because the relationship itself can be a bar to conversion. Some rabbis, mostly Orthodox but it's not unheard of in other denominations, will say that someone who is dating a Jewish partner isn't really sincere about becoming Jewish, they're "only" doing it to please their partner. And further, in an Orthodox context where it's forbidden for Jewish people to date non-Jews, you can get into a Catch-22 where if you were really sincere about being Jewish, you would do your best to keep Jewish laws, and that includes not dating someone Jewish when you yourself are not Jewish.

To be clear that's still not good; he's essentially asking LW to lie for him and approaching the conversion process from a starting place of dishonesty is a very good way to get turned down. The "refuse three times" thing is a bit of a myth, but there are stringent checks to make sure the potential convert is sincere, and lying about your motivations in that context is a big red flag.