I think, yes , you should.
There are cards for everything these days. Even blank cards that you can write your own message in. She's been around 3 years (recycled) so even if it's just a "thinking of you on your birthday" kind of thing, it's a nice sentiment. And, if nothing else will give you a bit of peace of mind... .
We often don't like what our BPDs do, in fact sometimes they kill the love and passion we feel for them; but we nons still have hearts that beat, minds that care and consciences that make us wonder "could i have taken the high road?... . on that"
So send her a card and stand on top of the mountain, it'll make you feel better
even if you are thinking to go down the other side to see if the other valley is greener... .