OK...let's revisit this one.
I've never had a problem such as you describe in an Eagle. This is what I'm thinking...
There is so little extra room in the transfer case that I think it would be almost impossible for the chain to jump over a tooth without showing some outward sign that it happened. In other words, I would think that if the chain is that loose it would break out the case.
You say you installed a new transmission mount, so I would not think that's the problem. That's one of the easy ones to check...the others being u-joints and slip yokes, which you also have done. Don't forget engine mounts. They're easy to check...open the hood, put a foot on the brake, place the transmission in gear and slowly accelerate the engine. Do this in "D" and "R" to test both mounts. The engine is visible from behind the steering wheel and will rock excessively to one side if a mount is broken. A good mount will give a little bit...I'm talking about rocking way over, maybe enough for the fan to hit the shroud.
A more likely cause, at least as I'm thinking now, is a broken ring and/or pinion gear tooth in one of the axles. It's easy to take the cover off the rear axle and inspect. It's a bit more difficult to take off the front axle cover because there isn't a whole lot of extra room. If it's just one tooth you may not hear a noise until you try to accelerate hard.