The Shulchan Aruch itself calls rejecting gifts a Midas Chassidus (i.e. not an obligation). If you have a choice between fulfilling something "extra" and fulfilling an obligation, you should definitely do the obligation.
Regarding the prohibition of offending someone, much ink has been spilled regarding the obligations of Chesed and Veahavta Lereiacha Kamocha (although some are of the opinion that it may not be a Biblical transgression of Veahavta Lereiacha Kamocha by passively not acting, per your comment). One such data point would be Sefer Ahavas Chesed by the Chafetz Chaim, Part 3 Chapter 8. That chapter deals with doing Chesed solely through words, and the fourth example given is lifting the spirits of someone who is sad through kind words. I think it would be correct to say that if it's a Mitzvah to raise someone's spirits, it would definitely be forbidden to hurt someone's feelings.
As an aside, looking at the words of the Shulchan Aruch (and commentaries), one could easily argue that the "gifts" it is referring to are gifts that one bases their sustenance and livelihood on (not just a "birthday present" someone gives you). While I would assume that it applies to any freely given gift, I wouldn't be shocked if someone pointed out a Posek saying that it only applies to certain types of gifts.