Personally, I would not change the differentials (you are aware, I hope, that you have to change BOTH of them on a 4X4!). That 2 or 3 thousand dollars would be very nice to have on the trip!
You should be fine with your 3.73 ratios. I towed over the major mountain passes (on I-90, I-80, and I-15) in the Northwest with a 1994 Dodge 3500 CTD with a 4.10 ratio rear end. I still had to shift down frequently. My son towed the same trailer over some of the same passes with a 3/4 ton Dodge with 3.54 gearing, shifting frequently.
Shifting gears is a fact of life when towing in the mountains. There is no gearing that will get over Lookout or Snoqualmie without shifting!
Have a good trip.
For comparison, I drove a 2 1/2 ton truck over a pass in Hawaii, with 1200 gallons of Avgas aboard, with 5.88 gears, and still had to shift down frequently.