The closest I found is 'psychological thriller', but it's much more on the psychological side, though it does have some thriller elements on the surface. But under the surface, it's about an unreliable narrator who suffers from depression, but hides it from himself and the reader, because that's what "real men" do.

You can definitely make that work. IMO-- and take this advice with the huge grain of salt it needs-- you should classify it as a psychological thriller but in your query explain just what you told us, that it is told through the lens of an unreliable narrator and explores mental health issues and how they cross with society's harmful view of ignoring treatment.

I haven't heard any agents say they won't look at something without a genre, but I think you'd have a lot of difficulty getting an agent interested if you won't say what your genre is. A lot of them specifically ask for it, and it's a bad look to not follow directions. Plus, even if it's true, I think it hurts you to say that your book doesn't fit into a genre.