Interestingly, that avatar doesn't use the exploit. After some fucking around, it seems to be something to do with the compression used in animated gifs, in particular only writing information that has changed since the previous frame. In Bizz-ness' avatar, only the tail is moving, which takes up less than 100x100 pixels (barely). I would assume that the "fix" involves finding the last instance of 0x21,0xF9 in the file, and checking the size of the frame following that, which ends up being the compressed frame.
This one will also upload fine. While the entire thing changes for the first 4 frames, only the corner changes in the final frame. Since those changes occupy a space smaller than 100x100, it passes. checking the first frame instead should fix that, since the first frame is always the same size as the entire gif (presumably).