I've never tried goo gone but twice now I've put what I thot was red colored glass in the dishwasher but it wasn't colored glass but some sort of colored coating over white/green glass. This coating turns cloudy in the DW and I've had to scrape it off. On this last one DH used paint remover which worked fairly well except for the crevasses.

I really need to look closely at cheap YS colored glass before I buy it.

Yes, the red color is most often painted on clear glass. I think it must be hard or more expensive to make a true red glass.
I do hope you can salvage the piece.

I would think goo gone, or an acitate based nail polish remover? Be sure to post your remedy when you find one.