Scott--
Bob is correct that you will not be able to evaluate the condition of the hub bearings until you extract the axles from the center section as you are now feeling all the friction from the diff internals and both hubs.
The work I described a number of posts back re resleeving the axle stub was not done on either of my Healeys but rather to my Elva Courier which uses a Morris Minor/Riley rear axle, identical to a Spridget axle housing only a few inches wider, probably the same as a Big Healey in 2/3 scale.
Though you don't have to remove the hub bearing in order to remove and replace the lip seal you do have to remove the hub from the axle housing to get a look at the bearing surface and lip seal. The method Bob describes is the same as what I do--in fact I have an old flange with an axle shaft that I cut down to about 6" long and after installing the flange in the reversed position and running the wheel stud nuts down onto the studs and if necessary shimming the flange away from the end of the stub axle just a bit further with a few right-size washers and tightening the nuts just a bit further I now have a convenient handle that allows me to pull/wiggle the hub that last little bit and get it off of the housing. Just be careful not to bugger up the fine threads on the end of the axle tube or the wheel studs.
Now that you have the hub in your hand you can inspect the condition of both the lip seal and the machined surface on the axle tube where the rubber rides and hopefully provides a good seal, though again I'll bet you will find some pitting, scoring, etc. on the tube and some chewed up rubber and maybe even a dislodged or missing spring at the lip seal which is allowing the axle lube to seep into the brakes and onto the inside of the wheel/tire as you first described.
Though you don't have to remove the bearing to replace the lip seal as long as you've gone this far you really should inspect the bearing to make sure it is in good shape, that none of the surfaces or races are scored, etc. Any metal from the diff eventually finds it way via centrifugal force out into the hubs and tears up the bearings.
To get to the bearing simply pry out the seal with a screw driver--don't worry that you're going to bend it up in the process because odds are it is already defective. Once you get the seal out record the number or mike its dimensions. Though Bob says you will need a press to remove the bearing from the hub this might not be necessary. Clamp the hub into a bench vise and find a socket that will just drop down into the hub and with the ratchet drive end against the bearing face (and/or a drift) and a good ballpeen hammer and some patient whacking you might be able to get the bearing out. If this doesn't work a press will certainly do the job. (I just love that sharp metallic ping as the bearing releases from the hub--the sound of energy!) Replace or reinstall the bearing depending on what you find.
Now finally back to that lip seal! If the axle stub is okay and there is nothing more than surface corrosion but no pitting or scoring clean it with fine emory paper and install a new lip seal into the inner face of the hub. Just be patient and tap it carefully into place with a soft-faced hammer, first on one side, then the opposite, etc. Once you get it started it will work in easily and land home.
If the axle stub is at all scored or corroded it is time to install a redi-sleeve or speedi-sleeve, a very thin and precision made piece of aluminum that comes with an installation tool though you may need to improvise with a deeper one than provided--I wound up using a PVC slip-on pipe cap which cleared the axle stub and picked up the shoulder on the speedi-sleeve. The correct sleeve number that corresponds to the lip seal is usually printed on the lip seal box. CR (Chicago Rawhide) is a good and commonly available brand and any good parts store can order both pieces for you.
Installing the sleeve is not difficult but takes time and care--I destroyed one learning how to do it so don't feel bad if you screw it up, though the sleeves are about $40-50 a pop.
When you get all of this done and back together and spin over the hub it will feel like a new bride and that leak will be gone.
Let us know how it goes.