Hi,
I agree that it would by far be best to install a new seal. Otherwise it's extremely likely that any new set of brake shoes will be ruined by future leaks.
However, it's the outer seal that needs replacement most and that requires a hub puller to remove the hub from the axle, as you have discovered. These pullers are hard to find and expensive. Perhaps a local British car repair specialist or local Triumph club might have one.
An alternative is to take the hub and axle assembly to a local machine shop that has a large press (probably 30 or 40 ton would be needed) and have them remove the axle from the hub. This is possible because the axle and hub assembly can be removed from the car as a unit (as already described).
Removing the axle & hub also gives you access to the inner seal (inside the axle tube) that controls the gear oil inside the rear end, as well as keeping the hub bearing grease out of there. That inner seal is an easy replacement, highly recommended while things are apart.
At the machine shop, a big concern will be avoinding fatal damage to the hub while pressing it off the axle. It takes a lot of pressure and whoever is doing the work needs to be very, very careful not to bend the hub in the process (Carefully support the hub all the way around... but even then some get warped, so there is always a risk here).
The shop also needs to be very careful the axle doesn't pop out and land on the floor, or it will likely be damaged, too (leave the large nut partially threaded onto the axle to prevent it escaping and falling).
Once apart, it would probably be wise to replace the rear wheel bearings at the same time. And, look closely at the Woodruff keys that lock the axle and hub together, checking for any cracking, signs of slippage or other damage to the key or it's slot. Also inspect the axle where the seals' lips run to see if there is any wear or scoring there. If there is, a "Speedi-Sleeve" might be used to give a nice, smooth, fresh surface for the seal to run on (Moss sells these, I believe). Another method uses a special seal that has an offset lip (or maybe two lips) that runs on a fresh area of the axle, avoiding the worn area where the original seal lip ran, which would quickly ruin any standard new seal.
I'd suggest rebuilding both sides at the same time. If one seal is gone, the other is very likely to go soon, too. Also, I think it's best to replace the bearings on both sides at the same time, if you choose to do so (Personally I would... considering the effort needed to remove the hubs).
Yes, as mentioned there are shims behind the brake backing plate that control the end float of each axle and how they mesh with the spider gears in the differential. Keep the shims carefully together as "sets" for each side when you remove the axle & hub assemblies. But with new bearings installed it will be necessary to measure the end play with a dial gauge and probably adjust (I seem to recall .004 or .005", but consult a manual on this), after which the axles need to be "centralized" so that the same amount of mesh is on each side. The adjustment is made by changing the number and thickness of the shims. This dimension is critical enough that the service manual warns to take into account any fresh coat of paint used on the brake backing plate, for the slight thickness of the paint itself.
It's possible the rear hub was over greaased and that caused the leak and brake fouling you found. But, to be honest, it's not uncommon for those old seals just to start leaking on their own eventually (Heck, they might be 50 years old, if original!). So, it might also be that the previous owner(s) serviced things well and properly all these years and the seal simply wore out.
Sorry this has turned into a bigger project than you planned, but I hope this info helps!
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