I think I would keep this hand? It definitely doesn't have everything you want, but I'd be afraid to go to 6. Note that 3 Cage, 2 Crypt isn't a TON of hate, and Cage is probably better than Crypt, meaning the chances of getting equal-or-better-hate plus more early game action might be low? I'm a little squeamish when it comes to mulligans though, so hard to say.
There's a pretty straightforward (if labor intensive) way to solve this though. Because this is a turn-0 game state in an un-subtle matchup, it's easy enough to just simulate and rate a ton of 6 card hands. There's probably an app or script online to do it, but you can also do it manually and take notes. It should be boring, but not difficult, to get a hundred 6-card hands and count how often they would be better than this.
My instinct says this hand is a keeper ... but unlike many more complex situations in magic, we don't have to rely on instinct here.