Had a similar problem a year ago... L1-L4 instability, causing 2+ years of chronic, severe back pain.
The physio suggested I start doing big compound lifts with correct form. So after a month of back squats, deadlifts, planks, pushups and specific lumbar exercises, I was pain free.
So:
- Back squats;
- Deadlifts;
- Planks (single arm/single leg);
- Barbell roll-outs;
- Band-resisted push-ups;
- Single-leg hip-pops (pelvic thrust things that FreeFaller loves);
- SLDLs...
Making sure you get the 'correct' activation pattern (i.e. correct muscles) is actually the important thing, here, so do the whole "pinch an inch in from your pelvis" thing whilst lying on the floor.
The physio noticed I also had a tendency to hyper-extend from my lumber region as compensation for poor shoulder/thoracic mobility, so I had a bucketload of other mobility exercises to do, too.