I try to only transplant once. Once out of red cups/ starter pots, into their final 3gal pots. You don't have much space in that fridge so transplanting should probably hold off until you move your plants to your flower chamber (which can't be your small fridge; unless you have a walk-in or something). I'm curious as to what you plan to flower in; perhaps a shower??? Remember plants grow 2-3X their height once you flip to 12/12, so pick something w/ enough head room. GL man!
I transplant twice usually. solo to 2gals then to their final home in 5gal. I find that the root system is much more complex and spread out through the entire 5gal when I do it that way. I did a test solo directly to 5 gal VS the work up method I just described.
The result was wayyy more roots via work up. The roots mainly just grow straight down then along the inside of the pots, so i feel like doing baby step transplanting makes more root mass on the interior/middle of the soil.
As for the flower chamber I posted a few older pics of that space.. I plan on slowly taking a few plants out of veg, transplant to 3gal pots, then move to flower... Trying out a new tech this go around... But I will leave 4 in veg box to grow big and clone, then have only clones in the veg box and the rest inside the flower room, which by that time wont have too many plants (I hope)