Personally, I don't send birthday cards to friends. So I would not send one to my W.
But if I saw a friend on the street and remembered their birthday was recent, I'd give my best wishes. So I SHOULD do that with my W, because I have no further reason to not be her friend.
If your goal is to continue to not be your W's friend, then do nothing.
If your goal is to be your W's friend, be her friend. And do with her as you would with any other friend. If you would send another friend a card, then do so with her.
Just do so with no expectations. And what ever she got out of it, that's up to her...
Right now... doesn't look like she's coming back to you... so who cares if she thinks it's pursuit... what's she gonna do? leave you?
