As you can see, this is pretty trashy - the background goes transparent after the first few frames. Is there a way to get the background layer to apply to all the frames? (I could use the Paint Bucket, but that would take a few years off my life. :P)

I'd use Gimp, it works pretty well for GIF. You make every frame into a separate layer, then save as GIF. Easier to know what's trans and what isn't since it's a full Photoshop alternative.