May I ask the obvious questions?
Does the 11 y.o. want to go on this trip?
Does GGM want to go? And does she want the child to go with her?
Does the child know GGM well enough that she would be comfortable spending the whole time with her?
If the answer is yes to all the above, well, think of the tremendous gift it would be to GGM to have this precious time to spend with a precious child. Next year might be too late. The clock is ticking down.
To be considered: will the trip involve a lot of time spent on a bus? Would the child be okay with that? If she gets bored in that type of situation, what about sending a electronic toy with her to keep her occupied/soothed during the boring times?
Sometimes kids are "needy/whiny" with one family member but quite different with another.