Aha! Finally! I've figured out how to eliminate environmental shadows on the avatar, and I'm an idiot for not doing so sooner. It's so obvious.

Go to World > Environment Settings > Environment Editor (or just hit Ctrl-Alt-E, if you like that kind of thing). Click the Advanced Sky button. Go to the Lighting Tab. Turn the Sun/Moon Color to 0 all the way around to eliminate the shadows. If you don't want to eliminate them entirely, just turn them down until you're happy: the lower you go, the fewer shadows and highlights will be cast by the sun or moon.

That's it. You're done.

You may want to turn up the Ambient setting to make the scene brighter, or turn up the Scene Gamma. You may also want to go into Atmosphere tab and make sure the Distance Multiplier or Density Multiplier is set to 0 (one or the other, though you can do both if you really want) to keep things from being hazy. Lastly, if you have local lights on, you may want to go back into the Lighting tab and fiddle with the Sun and Moon Position slider, because the sun position determines how bright local lights will be. Local lighting will also add some shadows back onto your avatar.

There are a couple downsides to this technique, though. One, the sky itself will look pretty bad. Two, your scene will be pretty desaturated and may look washed out, though I'll experiment more to see if this can be worked around. For me, though, it at least beats the awful shadows cast by the avatar nose.

Here's a picture taken using these settings, with local lighting enabled and an array of lights pointed at my avatar. It came out a bit small, but hopefully gets the idea across.