All the seven nations are allowed to convert by Biblical law.
While the nesinim were not allowed to marry among the Jewish people because of the takanas of Yehoshua and David hamelech, having a "tradition" that one is a nasin would not be sufficient to actually state that the person is a member of that group. It would be like someone who might be Jewish but has no way of verifying that a member of his maternal line actually was Jewish.
Similarly we see as part of the discussion of the Mishnah in the third perek of Maseches Kiddushin about the child of a mamzer and a shifcha. Thus, if one has a child with a nonJewish woman, the discussion in kiddushim would seem to imply that the pegam would not continue.
RambamIssurei Biah 12:23
23 They are called Netinim, "the designated ones," for they were
designated for the service in the Sanctuary. David came and decreed
that they should never be allowed to marry among the Jewish people,
even at a time when the Sanctuary is no longer standing. This is
explicitly stated in Ezra [8:20]: "From the Netinim whom David and the
officers designated for the service of the Levites." From this, we see
that he did not make the matter dependent on the Sanctuary.59
However, we also see based on Halacha 25 involving the actual nonJewish nations that are forbidden.
Therefore in the present age, in all places, whenever a convert
converts, whether he be an Edomite, an Egyptian, an Ammonite, a
Moabite, a Kushite, or from any of the other nations, whether male or
female, he or she is permitted to marry among the Jewish people