I appreciate the suggestion, but that's exactly what I want to avoid having to do. Even something as simple as a 5-second animation of a character running across a screen would be tedious as hell. For each individual frame, I would have to grab the appropriate running frame and paste it in just the right place. In Flash, the same could be done with a movie clip walk cycle, two keyframes, and a tween. However, I specifically don't want to use Flash or any other vector-based animation utility. I want to avoid things like misaligned or out-of-proportion pixels.
pixel animation requires detailed tweaks to various parts of a figure to look proper, it's the type of thing where you can't really take any shortcuts
I don't think anyone has created a pixel animator that
doesn't use frame-by-frame animation for exactly that reason
think of it this way: have you ever tried to rotate a very small object in an image editor?

blech
in fact a few moderately thorough google searches for programs recommended by people who do this type of stuff for a living or just as a dedicated hobby yields hundreds of thousands of results for ones that do frame-by-frame editing and are recommended by multiple people, including
graphics gale,
jasc animation shop,
pixen if you're on iOS, and
pro motion, as well as the aforementioned GIMP and even photoshop
in all of the message board posts I sifted through not one of them mentioned the type of thing you're looking for, which leads me to believe it either doesn't exist or if it does no one wants to use it because the results are undesirable
Also have I mentioned that these forums are the worst place to ask for advice?
just because one of the few thousand active members here can't answer your incredibly specific question in under 6 hours doesn't make this "the worst place to ask for advice"