Based on the information out there, it doesn't look like you'll have any problem with 200F+ diff. heat during the break-in period.
Here's an exerpt from an article on diff. temps apparently done by SAE:
"
Further evidence of stress and increased temperatures during the differential break-in period is documented in a 2005 SAE paper entitled The Effect of Heavy Loads on Light Duty Vehicle Axle Operating Temperature. A light duty GM truck towing 14,000 pounds was driven from Orange County, Calif. to the Nevada state line. The test was conducted with both a new axle and a broken-in axle. Over level ground towing, oil temperature was measured at 230 degrees F in the new axle and 203 degrees F in the broken-in axle. Oil temperature over the most grueling portion of the trip, during which a maximum 6% grade was encountered, revealed the new axle was operating at 350 degrees F and the broken-in axle was operating at 300 degrees F. Laboratory dynamometer test results simulating a truck hauling a trailer provided similar results, with level ground towing temperatures recorded at 266 degrees F with the new axle and 194 degrees F with the broken-in axle and towing temperatures (at a 3.5% grade) recorded at 370 degrees F with the new axle and 295 degrees F with the broken-in axle." However, that looks like diff temps on light-duty truck (maybe 5000lbs) towing 14,000lbs. In your case, we are talking a 3000-4000lb Jeep though. Your unbroken-in axle is running 200F and their truck pulling 14000lb load with an unbroken-in axle is running 230F (presumably both on level ground). Your new gears are 4.88 which are said to raise temperatures. They are towing which is also said to do the same.
What is impressive is that their truck's diffs are said to be running 350F when towing uphill (Holy Cow
). And they were apparently doing this with standard gear oil (which supposedly starts exhbiting negative wear characteristics at 200F)! It looks like the gears, if not the gear fluid, have pretty (high) reasonable operating temp limits. So take a look at fluids -- the likely weak point.
Personally, I think you are making a good move if you run synthetic gear fluid, especially during the break in period
(AMSoil seems to even make a special synthetic break-in gear oil, specifically for this type of thing) -- go with Mobil 1, Redline, Amsoil, etc. The break down temp on these oils is apparently between 300-375F degrees, likely 100+ degrees more than your standard gear oil. Also, they will likely produce less gear friction meaning less heat, so you'll probably see temps drop right there. Don't off-road, tow, low range for 500-1000 miles. Then change the synth fluid between 500-3000 miles to clean out any gear particulates in the fluid.
