Divorce

It depends on how acrimonious things are. Since she left while you were away and is communicating only through email it doesn’t sound like she has good feelings about you.
If the email is sincere and would not be received as “poking the bear” I would say yes.
However, from your description it doesn’t seem like it would be well received. It doesn’t really matter the motivation behind your birthday wish, it only matters how she receives it.
In the normal course of exchanging emails, you could add a small birthday greeting but I wouldn’t do a separate email.
You clearly know her and the situation better than we do so do as you know best. I’m sorry for your situation. The way she left and is communicating tells me she is avoiding confrontation above all else. Good luck.

Thanks. Good advice.