Hi,
Ouch, sorry to hear about the breakage.
I agree with your mechanic, that this might not be a DIY job. But you might ask around some more locally before making a final decision. Many truck repair shops should be able to handle rebuilding that diff, although you may need to provide them with the parts and a manual showing TR specifications and special procedures.
While it's apart, it's almost surely worthwhile to go ahead and replace the bearings and seals, but it sounds like you might already be budgeting for that.
You didn't mention what year TR3 yours is. Don't very early TR3 (and TR2) have rather fragile axles? If that's what's on your car, it might be worth looking for a complete later axle assembly, used but in good contdition, as a replacement/upgrade (you probably know that TR4 won't fit). I'm not sure exactly when the stronger axle design was implemented.
There was also a change in bolt size used to attach the crown wheel to the carrier. It might be necessary to drill your carrier out to the later, larger size, if you you can't find a crown wheel that's tapped with the smaller TR3-size holes. A size change will mean getting 10 new bolts and lock washers, too. Early cars used tab-locking washers, but rebuilds and later cars use regular split lock washers (and probably Locktite).
Shop around for a crown wheel and pinion gear set. The prices advertised by the biggest U.S. TR vendors seem pretty darned high. Just be sure it's a matched set.
You'll have a choice of ratios, too, if you want to make a change. 3.7:1 was typically fitted with 4-speed, non-OD gearbox, and is the most commonly found as well as generally the least expensive. Overdrive cars were often fitted with 4.1:1, which are rather hard to find and usually more costly, but will help acceleration combined with any gearbox. Without an overdrive, highway cruising might call for some pretty high rpms.
Some TR6 were fitted with 3.45:1, which will make for more comfortable highway cruising rpms even with a non-OD 4-speed gearbox, but will cost a lot of acceleration. There are a variety of other possible ratios, but these three are the most common and probably most useful.
If you aren't sure what you have in the car now, there are various ways to tell. Probably the easiest with the diff cover already off is to just count the teeth on the crown wheel/pinion. Pinion should have 10 teeth. CW will most likely be either 37 tooth (3.7:1) or 41 tooth (4.1:1).
A few other tips...
If you go the rebuild route, while it's all apart is a good time to check that the diff vent is clear. A shop will probably do this anyway, but you might ask just to be sure. A plugged up vent can cause leaking seals, and it's tricky to clean the vent on a fully assmebled axle without accidentally dropping dirt and grit inside the unit.
While the car is apart is also an ideal time to replace driveshaft u-joints and have the driveshaft balanced. If you do this, just try to avoid the u-joints that don't have a grease zerk fitted.
At the drive shaft, be sure to use new lock nuts on the drive shaft, especially don't reuse old Nylocks. I prefer to replace the bolts, too, *and* use all-metal self-locking nuts (such as jetnuts) *and* use Locktite. TRF sells a good kit of all the necessary hardware, if that's easier.
If they are still good, the outer axle bearings and seals might be left alone and/or done later as a separate project. They can be removed and worked on easily while the diff unit stays in the car. Just be a little careful removing and reinstalling the axle shafts themselves, to not damage the lip seal inside the axle tube. Watch for the shims, too, that control axle end-float. You will find them between the brake backing plate and the axle tube flange on each side. Mark and keep each set of shims together for reinstallation, but expect to have to check and reset axle end float (by changing those shim packs) after having the diff work done.
Best of luck with your repair!