It’s understandable how you feel, and there are probably many others who have wondered the same thing.
It sounds as if you have already thought this through and considered the pros and cons from your ex’s perspective as well as yours. If there is any reason to think that the ex might consider a birthday wish from you as either a slight or a sign of hope (albeit false), then for the ex’s sake, it is better not to do anything at all. If the break-up was amicable, then surely the ex knows you well enough to read this as one more kind act that you normally do based upon your personality.
If there are children involved, encourage them to make or purchase cards or presents for the other parent and consider that your contribution.