I thought this was an interesting exercise so here is my attempt. OpenG has some image tools and one of them is the ability to open a GIF, but for some reason it crapped out and died with your GIF even after resaving it to something much smaller. I did find some other GIF API over on the dark side and instead used that. Attached is a zip, extract it and run Demo Saving Button. It will show the first image. Then when you click the image it cycles through the first half of the GIF and waits for the simulated save process to complete. Once it is complete it rotates through the second half of the images, and then after a few seconds returns back to the first. Parsing of the GIF takes time so I put in the GIF images as a constant, along with the code to parse the GIF. I also set the pane to be the color of the (0,0) pixel in the hopes it will blend in better. Honestly this could be turned into a QControl and be made very seemless.
Demo Saving Button Gif.zip