A.E.
My kids birthdays border the holidays as well and we have the same issue. We have a couple of different approaches--one Grandma gives them their big birthday gift on their half-birthday -- in the summertime when they can get something they couldn't use in the winter such as a trike, inflatable pool, bike, scooter, etc.
Another option we've done with a close friend whose daughter has everything and birthday is shortly after Christmas, is to give her a gift card along with something small for her birthday, The gift card is for an outing for her and her Mom to join us -- to a movie, trip to Color Me Mine, play, or for a boy it could be Lego Discovery, etc.
AE