It is clear that we are not obligated to follow the Minhag hamukem if the minhag has no halachic basis. For example if my Bubbe ate chicken soup on Shabbos that is not binding upon me, since it has no Halachic basis.
The way i see it, the minhagim of sefira have no halachic basis either. It is not mentioned in the Talmud (though the story is mentioned) and something that is not mentioned in the Talmud can never become part of Halacha binding upon all Jews. Furthermore there's no Halacha that a Jew is commanded to mourn the death of Torah scholars. Can we say that these minhagim are not binding but optional?
On the other hand, the Tur and Shulchan did incorporate them into their Halachic code (though the rambam did not). This would seem to suggest that they are binding upon all!
P.s. This question can also be viewed as a general question as to what makes a minhag binding.