I say, "Celebrate them Both!"
With everyone being so busy these days sometimes it's just too hard to celebrate an event on the actual day.
DH and I had our "honeymoon" trip to Disney two months after our wedding day.
Last year we celebrated DD9's birthday four months after the actual day. It was the only time we could get to Disney. We tend to always travel during MY birthday and DD wanted to celebrate HERS one time at Disney. So, I gave her MY birthday. She wore my birthday pin and accepted all the "Happy Birthdays" from everyone. Such a simple thing, but it made her so happy and feel extra special. (She even wore her "birthday" hat from Whispering Canyon onto the airplane for our trip home. A flight attendent took it up front and showed the pilot and then had everyone sing Happy Birthday to her!

) It really was an extra magical trip for her
And if you're worried about "cheating" Disney out of something, I don't really think you are. We had paid for the dining plan. Whenever DD9 was brought a birthday cupcake the dessert which she had ordered was never brought! So, the dining plan paid for her "free" cupcake.
You'll have to decide how comfortable you'll feel when DS9 is asked, "Oh, is TODAY your birthday?" My DD replied with, "No, but we're here celebrating it
this week." Never heard a bad comment about it.