It's actually not common at all to have 5 years left to live. A few months, a year or two, or "there's nothing we can do" are the usual diagnoses. That said, whether a person is dying or not, their birthday is a time to say "I am glad you haven't died yet" in one way or another, and so it would be proper and good to celebrate their birthday, knowing it may be one of the few remaining ones.
Yes, it will be bittersweet. For you, and for Alice. Same when you celebrate your child's teen birthdays knowing that soon the child won't live with you, won't want balloons and cake, may not even visit near their birthday. Any time you step aside to mark the passage of years and the long rhythms of time, you look forward as well as back, and you acknowledge that what you have now, you won't always have. That's kind of the point.
And finally, as someone who was dying (I got better) one of the terrible things about it is people making decisions for you and taking good things away from you because "it would just upset them" or "it might remind them they don't have long" (as though somehow we might forget without your invitations and good wishes.) Wish Alice Happy Birthday with all your heart. Tell her you love her, she's an inspiration, you want to be like her when you grow up, and that you hope you get dozens more birthdays with her, and that if you don't, you're damned grateful for the ones you've had. Cry if you need to. Then smile.