My health deparment requires a written procedure for handling the chicken in the cooling stage, but it is pretty simple. We do exactly as you mention with the fresh wings . . . 7 minutes at 350F and check internal temp (I’ve never been below 160F. Then I cool in pizza dough boxes on counter cross stacked for about 10 minutes, then transfer to the reefer cross-stacked for 30 minutes until below 65F, then into lided plastic tubs boxes to finish the cooling process.
I tested temps in the tubs in my shop for temp drop over 2 hours, and all fits nicely into published cooling time/temp ranges. It sounds way more complicated than it is. We use slack times early in the week to process wings for the whole week. We have never been caught short, because we replace what we re-fry during slack fryer times each day. Some shops just parfry every day for the day’s load.
LABEL YOUR BOXES so you don’t lose track of which day they are processed.