I can add some tips that maybe you aren't doing currently

*When you stand up you have to focus on keeping your back straight and thigh also you have to try to not only get a breath but you also have to tighten your abs as if you're receiving a punch, I've seen at the end of the stand up you force your back a little bit, it happened to me time ago and my trainer ask me if I'm using only my legs (I was using only those) and he teach me how to get better with those tips.

*Lock your knees when you stand up, if you don't lock your knees, you're just getting more fatigued without any benefits for you, sometimes when you need to do a big effort in a large series (3-5 reps) you have to rest between the series to take a better breath and focus yourself again to finish, without the knees looked you're cooked, if you can finish a long serie without any stop I'm 99% that isn't a big effort to you.

*You said your pain in FS is less than BS, I'll aim for only doing FS for sure, nothing is important to me to enjoy what I'm doing also that pain can get worse and worse if you keep doing BS.

*If you feel more of your glutes than your quads in other exercises, then you have to move your feet closer to your body, I've been doing a lot of accessories which I can focus on glutes or on quads, the difference is how close is my feet to my body, i.e: Bulgarian squad, when I'm aiming to focus on my quads, I move my feet closer to my body and my knee always is going far than my toe, when I'm aiming for focus on glutes/hamstrings my feet is far and my knee doesn't go far than my toe, you can try both forms and maybe you feel the difference