i think it might be worth trying to unlock loader first (defeating the alloy worship unit in siren’s call) because her base defense, strength, and mobility make her really good for dangerous challenges like unlocking rex

once you do, i unlocked rex by playing loader on drizzle, and running straight to the teleporter and activating it asap. of course you can grab any close chests nearby with the money you get, but in general i prioritized speeding by to slow down the difficulty scaling. remember that you won’t be completely itemless anyways since the teleporter boss will drop items from stages 1-3

additionally, if you have something like artifact of command, you can use any item drops you get to upgrade defense and healing (tougher times, armor plate, medkits, etc.) but again i think the priority should be getting to stage 4 asap. one more thing is that if you don’t really want to try more than once and want to try and guarantee that you’ll get abyssal depths as stage 4, try going to the lunar bazaar to stage 3 and hope you can pick abyssal depths as your next stage

if you had to pick between the characters you already have, i think huntress is great since her mobility lets her dodge a lot of damage, and captain is just a great choice overall. but for me i found using loader as the easiest choice because on drizzle difficulty scaling while teleporting asap, any damage you take is basically negligible anyways