Right now, as a Republican, he is providing a veneer of bipartisanship which is politically effective when it comes to getting the Epstein files released. That is to say, it looks much better from the point of view of people who don't follow politics if the charge to release the files is being led by congresspeople from both parties -- it makes it harder for TrumpCo to pass it off as a "partisan witch hunt," which is what they would likely attempt if Massie switched parties. (Plus, Massie isn't going to switch parties, because at the end of the day he's a Republican and he doesn't agree without our principles.)
Similarly, it benefits the Epstein investigation if Massie remains in the House. If he resigns, it's a symbolic gesture, but he would then have no power. He would be replaced (presumably) by another Republican, who would likely be a MAGA supporter. Bear in mind that it was Massie who helped push the discharge petition over the line which got Congress to pass the law demanding the release of the files. Only four Republicans voted for the discharge petition -- Massie, MTG, Nancy Mace, and Lauren Boebert -- but that was enough for Democrats to force the issue, which resulted in all but one House Republican voting for the release of the files.
Without a majority in the House, Democrats have essentially no power, and having a handful of rogue Republicans on the other side is what's allowing them to make these procedural moves and give Mike Johnson a major headache.