Meat processing in winter isn't needed, just push the entire pile (-1) to nearest tree or boulder to keep wild life and dogs from eating it.
It'll preserve fine until spring.
I'm not sure why there's 2nd sleep in your scenario, unless woken up by rain/noise before Vigorous. eating a Bearpipe before passing out at "ready to drop" will speed the sleep duration from ~12h to mere ~2-3h (takes 1/2h to kick in, boost to Lively, then sleep to vigorous. e.g. Ready to Drop at Little Hours, eat bearpipe, lay down to sleep. Wake up Early morning-Morning at Vigorous.
Yeah sadly Bark isn't harvestable in the cold/frozen weather. It should be doable to get bark scraped off and boiled into tanning liquid.
I try to harvest tanning and crafting bark in the early summer after "stabilizing" from spring start, before plants are ready for harvest. (time it with cabin build et cetera) 30-40 pieces should do in the few cases when already "Extremely tired" when making the kill.
Also trying to time the skinning to finish just after day-tick at Morning, or well before to have time to clean&applying tanning agent.
fresh killed carcass hasn't ever lost 'condition level' for me over the day-ticking over, but sometimes, the skin might.
Still, I do agree to have the spoilage timers checked on tanning/de-hairing large skins.
IRL, when I don't have time/interest/energy to work on skins, (from hunting) I lay them between newspaper sheets and roll up. Put them in freezer for later processing. No noticeable deterioration in at least 4months. i.e. winter time skin shouldn't rot. except when left too long with tanning agent on them. Fresh skin, clean skin, rinsed skin all should "stay good until spring"