You're really gonna have to lose alot of frames. Even at avatar size it's 1.2MB.
Only reason I was able to resize your current one was it's a very very simple gif. I can try and get you a working version, but it won't be the full loop unfortunately.
It has 175 frames. It's impossible to make it 125kb or under like that
I'll play around and give you a few options.
This was the first resize. Only resizing the dimensions so it fits that criteria. It's 1.2MB so isn't avatar worthy on this forum.
The second I cut it down from 175 frames to 50. It's still pretty big in size, and too big for an avatar.
I lowered the quality for this one. It's still too big but not greatly so, maybe PM'ing an admin and asking the 125kb limit to be raised to 200-250 ?
Only other options are to lower the amount of frames again, and maybe even lower the quality more (pointless, it looks like crap)