I've run 4awg front to back on two 5th gens. There's room on both the driver and passenger side. I'd imagine 2awg would fit but I would check first, that cable gets thick fast.
Running the cable through the firewall isn't an issue. I like to use dynamat or the like to seal any holes made through the firewall, even if it's through one of the factory rubber grommets.
The tight spots will be under the door trim of the front and second rows. I'm not sure if you've looked at this yet, but the trim is held in with pressure clips, simply lift up gently and it will pop out. There are factory wiring looms running down both sides but there's extra space for accessory circuits. As I said, I know 4awg fits, not sure about 2awg but I'm thinking it will.
You may have to trim some plastic to get from the floor of the second row to the rear cargo area, depending on how you route the cable.
There are a number of ways to access the undercarriage once you get the cable to the back of the truck. I chose to drill through where the cargo rings are at the very rear of the vehicle. This was due to the fact that I mounted a rear fuse/relay box back there. For me it made the most sense, my wiring is enclosed in a durable pelican box so gear can be stored on top, as opposed to having the wiring running up the side of the vehicle. I power several things out the back though. You could also pop the plastic trim on the floor of the very rear of the truck and route the cable under the floor of the cargo area of the vehicle. I'm not sure if there are any factory holes to run the cable to the underside of the vehicle there though, you'd have to check.
Either way, you have direct access to the hitch under the factory cargo rings, so it's an option.
Overall it'd be a lot of effort and $$ to get a cable that large back there. Practically speaking I'd probably ask myself how often I'd be using a rear mounted winch, and whether or not there are other solutions. Hope that helps.
Edit: Just saw you asked about under the vehicle, not inside. Due to risk of insulation damage to the cable I'd probably stick with your initial plan for an exterior extension cable. I suppose you could always go along the frame using zip-ties or cable-clamps, seems prone to issues after awhile though.