Harbor Freight or Lowes/HomeDepot bottle jack will work just fine if you have room for it. Get under the jack point and find out how high you need the minimum specific to your tire and lift combo
Like previously mentioned, make sure the collapsed height of the jack will fit under your jack point, measure as if you have a flat (the truck will be a tad lower
Oh yeah, I already thought of that but thanks for the reminder...both of you
Your jack will probably be stored under the rear seat, so I hope it is a tad above -20 inside there.
Nuthin works on me below about 0 degs F, so I don't see worrying about the jack.
It never fails, I would find myself with a flat and sitting in mud that made my vehicle sink further.
If you are going to go this far with it, consider a block of wood, maybe a piece of a 2x6, to place under the bottle jack so it does not sink into the mud. Most bottle jacks have a pretty small footprint.
That's assuming the bottle jack does fit under the seat???
Sound advice about a bigger footing for the jack and that additional height has to be taken into consideration as well. Also, something to block a tire with...