The gemara discusses exactly how much of an animal must be given over to a goy in order to avoid kedushas bechor. The predominant opinion seems to be like R. Huna that the ear is enough, since we find that Rami Bar Rachel did that as well. Tosafos entertains the possibility that we would pasken that one would have to give over to the goy a portion in the animal that if missing would render the animal a neveila or treifa (rav chisda and rava), but considers it to be a chumra. Regarding the fact that one is removing kedusha from animal entirely by selling a portion to a goy, Tosafos suggests that it would only be an issue if one were to sell a portion of the fetus which itself would have otherwise had kedusha, but would not be an issue when selling a portion of the mother. Even though this may not be ideal, Tosafos considers it the proper thing to do nowadays since there is no better alternative and if this isn't done someone will surely violate a more severe prohibition with the animal.
The Turei Even (Rosh HaShana 13) suggests that being mafkir the animal would NOT be sufficient to remove the kedusha of bechor from the fact that Rami Bar Rachel who was looking for the simplest option, didn't use the method of hefker. The Chasam Sofer (Y.D. 316) dismisses the proof since it could be that a proper hefker in the presence of others could have been more complicated and difficult than simply selling the ear to a goy. The Maharit Algazi suggests a complicated approach. Hefker wouldn't work to remove the obligation of bechor on an animal that already has kedushas bechor. However, if one were to be mafkir the fetus prior to it's birth, since at the time when it would be fit to receive kedushas bechor it doesn't have an owner, it wouldn't get kedushas bechor. He then backs off based on a gemara in Chulin and says that even if born as hefker, as soon as a Jew would take possession of it, it would assume kedushas bechor.