Yeah you don't necessarily have to do a lot of S&C on the side, but it can be useful. Your weaknesses and strengths would help determine what's right for you, but that's not possible to discern over the internet.
BW exercises do have their pros and doing squats like you are might have some benefit. Still, if you have access to a gym and is already devoting some time to doing exercises, I'd recommend putting perhaps barbell squats and trapbar deadlifts in there. Controlled reps and gradual buildup.
It doesn't have to be complicated, it could look something like a lower body compound exercise, an unilateral/single leg exercise, rope climbing/chinup variation and perhaps a core exercise like turkish getups. Just an example. Point is, it can be in small dosis and gradually building the load so that it doesn't interfere with your sport.
The plyos/jumps and all sounds great too, and the rings if you can handle them.
In regards to your weight, do you feel like you can control it somewhat by what you're eating? Usually the amount of calories you take in is the biggest factor in how much size you gain. That and the cardio you do might be something that would help you not get too big for your weightclass.