Pilates to strengthen your lower back?
You know what works best for lower back strength? Deadlifts.
Deads also also strengthen calves, hamstrings, quads, lower back, mid back, upper back, traps, neck, shoulders, grip/forearms and abs, including internal core muscles.
Pilates strengthens vaginas.
There is nothing wrong with doing super light deadlifts, especially if you are filming them and watching whats happening to your back and your setup. Add on top of that hypers, dumbbell rows (real ones, with the weight moved back to the hip joint, not up to the chest) and anything back focused. Hammy work such as leg curls, romanian deads, etc.
THAT will strengthen your back. Pre gym life I would put out my back putting groceries into the boot. Post gym life I can safely pull double body weight and more.
On the matter of timing for your workouts, you are a beginner so anything will work. But, bench hits mostly pecs and shoulders. Squats hit everything so a squat or deadlift gives you a much greater strength and growth potential.
Switch time spent on squats with bench, or increase squat time (as in the actual squats, not accessory exercises) to match bench. PLUS add in deads on their own day.
If your workouts are more of a mental challenge than physical, then you are doing it right. Mebbe throw in a deload week every 4-8 weeks to give yourself a physical and mental break. You should feel fear and anxiety facing a squat or deads day, but not to the detriment of enjoying the experience.
Hang in there. You are on the right track. Hit that posterior chain with light deads, hypers, reverse hypers, leg curls, romanian deads, etc and your lower back will strengthen up fast.