Most Helpful Guy
Texting him on his birthday would usually convey good will if the breakup went smooth and was mutual. Doing so that soon after separation does complicate it, especially if there is bitterness on either end - might elicit a more grudging thanks, but just because the friendliness is dissonant with current feelings.
In my experience it feels better to have someone who was in your closest confidence still friendly after separation. Wishing him a happy birthday sets you on this path, and the risk is better than the risk that he would have wanted such a wish and took note of that you didn't send it.
Most Helpful Girl
Well who broke up with who? Either way if it were me, I wouldn't. It's a waste of time. I mean, you just broke up. The most you'll probably get out of sending that text will probably be a dry ass, "thanks" and that'll be it.