I'd like another try at this. I noticed that you said you "...really have to squeeze...", which, if you are not running the lever into the front of the handlebars, suggests that you would be willing to trade off some travel for some leverage, and guess what, with cantilevers there is a way to do that (you do have cantilevers, right?).
If the wheels are true, keeping the pads next to the rim, if you shorten the straddle cable, you will give up some pad travel for some leverage, and, of course, vice-versa, you can lengthen the straddle cable to get more pad travel with the trade-off being that you give up some leverage. Could be that you can adjust this and solve your problem.
Four cautions:
1. You have to keep the straddle-cable-saddle-thingie at least low enough so that it can never run into the bottom of the cable housing stop. There is a slim chance that is the source of your current troubles.
2. With all cantilever systems, you have to be sure the system is reliable enough so that it will never drop the straddle cable onto the wheel.
3. Make sure you can't run the lever into the handlebars with your new setup.
4. It's been so long since I've done this that I am a little fuzzy on whether to lengthen or shorten the straddle cable. Think I got it right.
Down here in windy North Texas, it is dang-blame hard to find anything that anybody from any other part of the country would recognize as a hill.....the nearest big hill (ahem) has a 350 ft elevation gain, and I think the nearest 400-footer is out by Palo Pinto (HWY 4), about 90 miles away.
I don't know much about hands getting tired braking......brake pads do tend to last a very long time here.....