Squat alternative.

I would guess it's a toss up between hack and belt squat. But assuming your gym has dumbbells, you'd probably get more carryover from a heavy goblet squat.

strange gym that doesnt have a squat rack but has a belt squat

that makes sense, your gym probably has a landmine attachment somewhere so thats an alternative or cable squats are another one

I would do DB lunges.

Have you access to a sissy squat?

I'm no physio but id probably suggest seated leg extension to grow your quads and keep your back out of it. That type of isolation work would be better suited to you if you're minding your back.