If the vibration is only present under load as you described, then it’s in the driveline FER SURE. If it were a tire, front end component, axle, driveshaft, etc, then the vibration would be consistent regardless of load.
Since the only thing that is different from what you’ve described, is the the differential/ring and pinion. My guess is the backlash is too tight. A cheap, and relatively easy test is to increase the backlash by .002 - .004”. Pull the axles, then pull out the differential/ring gear and move .003-.006” of shim from the left side and put it into the right.
Gear setup is like rod and main bearings clearance - a little too loose and no one will know, but too tight and everyone (and you) will know. I run the backlash at the top of the spec or .001-.002” too much. The gears will be fine a little, repeat a little, too loose. Considering I’ve been doing this in the 800-1000hp range, I’m sure you’ll be fine. If you feel that this adjustment is over your head, but you can do basic repairs, then you’re wrong. The only tool needed is a budget caliper. The only parts needed are a rear shim kit (less than $50) and some silicone.