The Mishnah here does not discuss the subject you asked about. The rights of a Bechor with a Mum are explained elsewhere, and there is no doubt that the Bechor should be given to a Kohen. The Mishna here discusses the laws of seeing Mumim on Yom Tov, and whether a Bechor which was not inspected before Yom Tov is Muktzah. It is possible that this is a Bechor that was given already to the Kohen and then fell into a pit, or alternatively, it is still by the Yisrael who wants to save the Kohen's money and now wants to raise the Bechor from the pit so that the Kohen does not lose.

There is a broad issue that arises from the Gemara (Baba Kama 11B) what is the law of a person or an ox that harms a Bechor. This depends on the question of whether a Bechor with a Mum is considered "Mamon Gavoah" and belongs in some way to Hekdesh or a capital of the owners, but in any case, it appears from Rashi (Baba Kama 51A) that damage to a Bor (a pit) is in any case exempt from damages to a Bechor. Also, as long as the Yisrael did not give the Bechor to a Kohen, it is difficult to understand who from all Kohanim is the specific owner which the Yisrael will need to pay to, if he actually caused the Bechor to fall into the pit. For these reasons I did go into these aspects.

Aharon Steiner