Rude to dip grandmother's ring?

Your question asked if it would be rude to ask your grandmother etc... My answer is no it would not be rude. Before you ask her tell her to be completely honest with you and you'll respect her answer. Explain why you want to get it dipped and see what she has to say. If she is opposed to having it dipped and you really want to wear it then do so. If anyone ever inquires why your band and e-ring don't match you can explain it! I would have been ecstatic to have had my grandmother offer me her ring and would have worn it regardless if it matched or not but that's just me.