I am in the same boat regarding little sleep, except with kids not work. I squat way less than you and I started a Light day.
My rationale for it was that it took the mental tax of Wednesday’s squats off my shoulders and it gave me a little more recovery time, meaning Friday’s workout started getting better. It also got me excited to workout again.
I waited til I stalled once to add the LS day. To me it seems logical to add it when you’re averaging the progress with it anyway, I.e., when you should be doing 15 lbs a week but with resets and missed reps in reality you’re hitting 10 lbs a week.
Andy Baker once told me that if your form is “trainable”, meaning not so awful that it’s unsafe or ingraining bad movement patterns, you should continue to add weight. You don’t want perfection to become the enemy of progress. Note what’s wrong and work on it during your warm up sets. If you take a Light day, that’s also a good opportunity to practice technique.
I have no experience micro loading the squat. Just not there yet.