Well I've played some more since I've asked this question, and I think I can deal with heavy armor a bit better now. My conclusions:
First, even for a pure warrior with heavy strength investment, you can't just pick the best armor around. Instead, I check the armor-to-weight ratio of various armor pieces, and choose appropriately. So if I have two available upgrades for my boots, one weighing 3 pounds more and provides 1 more armor, and the second weighing 9 pounds more pounds and providing 2 more armor, I'd pick the first one.
It's true that if you look at a specific armor slot then it seems like you're missing on valuable armor points, but overall - if you keep upgrading to items with the best armor-to-weigh ratio, you will gradually improve your armor while not overburdening yourself.
Second, the ogre's strength effect, which can be obtained by casting the ogre's strength spell or from potions, is very useful for temporarily increasing your carrying capacity by a great margin. If you can maintain it over a long duration, good; otherwise, it's possible to exploit the fact that effect duration is not reduced by fast-travel. Just hoard all the treasure in one chest, and when you're ready to leave take it all back, cast ogre's strength and then fast-travel to the nearest city to sell.