Stumbled upon a way to use animated icons within the Self Service app - Simply use an animated PNG file (APNG format). Looks like support has been around since iOS 8 and OS X 10.10. Have not tested this in Self Service for iOS.
This seems pretty harmless and if you stick to the recommended 128x128 canvas size, you should be OK. Maybe JAMF can comment :D
Below is an example I put together, two frames looping. This will animate if you use Safari or Firefox but not Chrome.

