Is the paint stable, or is there scattered flaking? My brother took apart an orange crush clock, tried to spot clean one dingy area, and 30 minutes later was staring at a clear piece of glass. I rebuilt an Iroquois beer double bubble about 5 years ago, that had dirt streaks on the inside that were very visible with the clock lit up. I used a paper towel and water, and was aggressively scrubbing by the time I finished. Not a spec of the image came off, and 5 years later she still looks great. So judge if it can be cleaned, then try with plain water to see if anything will come off, starting with the outermost area to test it.
Just thinking out loud here, but is there a chance the glass side is dirty vs the image side??
Hope this helps.
