Lower Back Injury and Programming for Rehab?

WARNING: LONG POST!
For a while now (actually, since November last year), I’ve been having recurring issues with my lower back after performing squats, deadlifts and power cleans. Basically every exercise that somehow involves the lower back seems to trigger the issue. It started out of nowhere, after a certain training session I got this almost constant burning sensation in my lower back. It got more obvious while lying down as well. The ‘pain’ was more annoying than it was actually painful, but it concerned me enough to go see the doctor. She told me it was probably some sort of inflammation (not a hernia as I had feared). She prescribed me anti-inflammatory pills called ‘meloxicam EG’ and sent me off to physio. The physiotherapist bent me in all kinds of ways and said he couldn’t really pinpoint the problem. He did mention my glutes had some tight pressure points. After that I started foam rolling my glutes almost daily and it helped somewhat in relieving the burning sensation, yet it kept coming back after x amount of workout sessions.
So basically, ever since November last year this problem kept coming back and I don’t really know what to do anymore. What I’ve gathered from researching this, it might be my weak core/glute activation that is causing this issue. I’ve always been struggling with the squat, power clean and deadlift, despite reading a lot about it, watching many technique videos, filming myself and trying to be as ‘aware’ as possible. I never really feel ‘strong’ at any point while doing these exercises… I haven’t targeted my core in years, since I read everywhere that squats and deadlifts provide sufficient core work… To give you an idea of my core strength: a simple plank lasting 50 seconds has me shaking all over the place and is actually hard…
My plan was now, for the following month(s), to omit the squat, power clean and deadlift from my sessions and instead focus on getting my core and glutes much stronger. I figured I can keep doing upper body stuff as it is, except for BB rows and BB OHP, since those two involve the lower back quite a bit as well. How would you guys go about this? Which exercises do you recommend and how would you program them? I’d have to focus on the abdominals, lower back and glutes. I have barbells, dumbbells, kettlebells, a power rack and two benches (flat and incline) at my disposal, no other machines. I also don’t want to lose any quad/posterior chain strength, so maybe there are some lower back friendly ways to train these as well…
The goal is simply to get rid of this low back issue and to get back at squatting and deadlifting (safely!) as soon as possible

Thanks in advance!