Are there any tests I can perform on the transfer case? I don't detect any slop when I try and turn the pinions by hand.
Unfortunately, I don't know of any. The gearbox is just that: a box with gears inside. They are all mounted on shafts, which are supported by bearings. Really, only three things can go wrong:
» 1. seals fail, gear oil drains out and the gears eat themselves
» 2. bearings fail and the gears eat themselves
» 3. gears fail and the gears eat themselves
If there's a bind, as in a shaft that's not parallel, in the geartrain that is the result of a failing bearing, it might show itself as a binding that resolves itself after the shaft starts turning. I've never seen this happen, but it's possible.
There might be a problem associated with the viscous coupling, but offhand I can't think of what it would be.
The transfer cases are really simple assemblies, mechanically, so it's odd that the problem isn't more obvious.
Starting from the front of the drivetrain and working back:
» You can eliminate engine–transmission mounting related probelms (I assume you've checked the bellhousing bolts), because the two are actually one unit once they're bolted together. Nothing there would "suddenly" present itself as a problem. If there was a misalignment, it would start bad and get worse, not start bad and go away.
» Nothing within the transmission manual shifting mechanism would present itself as a problem as you've described, which would resolve itself completely when the transfer case is positioned in neutral. That mechanism is just a rod and lever; you've seen it when you removed the pan. If there was a problem there, it would have been obvious.
» Removing the driveshafts isolates the transfer case from the remainder of the drivetrain. If the problem persists when the driveshafts are removed and the transfer case is engaged, but goes away when the transfer case is disengaged, the problem is in the case.
» It's unlikely, at least in my mind, that the mouting between the transmission and transfer is to blame, but that's easy to check. If all the mounting bolts are tight, the two are correctly mounted. Unless there's a crack in the case...which I would assume you'd see, or gear oil leaking would show you.
Unfortunately, this is a problem that takes eyes and that's hard to do from where I am. I wish I could be more help.