First off, have you had your back evaluated by a professional? If not, then if at all possible with your family history, I would prioritize that. This may help you be able to decide how cautious you actually need to be.
I'm not sure why you would want to omit training any part of your lower body or back to avoid spinal issues. This would actually more likely lead to injury on the mat. There is a way to work every leg and back muscle with limited spinal pressure. Having muscle and strength will actually help prevent injury on the mat. It's a form of armor building and can build health movement patterns. Also, if you're totally outclassed in strength with similar skill with your opponent it's not hard to imagine that you would be more vulnerable than if your strength levels were equal or higher.
In fact, not training your glutes can lead to back problems in sport and every day life.
Now before you totally write squats off, there are squat variations that don't put a lot of strain on the lower back. Here are some examples:
*Front Squats- even for people with healthy backs, this can be a bread and butter lower body builder for nearly anyone instead of back squats.
*Goblet Squats- extremely limited back strain. The biggest problem is that you can't use much of a load, but if hypertrophy is the goal many people respond best to rep ranges 20 and higher for quads
*barbell hack squat
*Manta Ray Squat with Raised Heels- The manta ray attachment essentially makes your a back squat a 'higher high bar' squat. This keeps you more upright.
All of those options are more quad dominant so you'll need to do more work than many people working on your posterior chain with other exercises.
Here's back friendly examples:
-glute ham raises
-hyper extensions- the only thing hyperextending is your pelvis. Don't hyperextend your back.
-cable pull throughs- wider stance is more glutes and hips and narrow stance is more hams and back
-dumbbell romanian deadlifts- much more back friendly than its barbell cousin
-1 legged 1 dumbbell that's opposite hand of the leg your standing on- this can help prevent back problems
-bridges- be careful not to hyperextend your back
-thrusters- basically bridge with your upper back on a bench
There's many more, but that's way more than enough for a start.