Pesachim 89A

משום דאיכא חזה ושוק, דכהנים הוא דאכלי ליה.

because there are the breast and the hind leg, which the priests eat only if the offering is a peace-offering, but not if it is a Paschal lamb. Due to the distinction between the two types of offerings, the proposed stipulation does not provide a solution. It would remain unclear who should eat these portions of the offering.

וניתי כל חד וחד כהן בהדיה.

The Talmud suggests that it may still be possible to find a way to use the stipulation: Let each one of these 5 people bringing a new Paschal lamb bring a priest to be registered with him for their Paschal lamb. Then the priest will eat the breasts and hind legs. Since the priest eats these portions of the offering in any event, there is no longer any practical distinction between the two types of offerings. The stipulation should therefore provide a solution.

האי כהן היכי דמי? אי דעביד פסח ⁠— דילמא האי פסח הוא, ונמצא פסח נאכל שלא למנויו. ואי דלא עביד פסח ⁠— דילמא שלמים הוא, ולא עביד פסח!

The Talmud rejects this suggestion: What are the circumstances of this priest? If he had already performed the ritual of the Paschal lamb during the first Pesaḥ, then, as the Talmud explained previously, he is unable to be registered to partake of a new Paschal lamb. If so, there is concern that perhaps this offering is a Paschal lamb and not a peace-offering, and it will turn out that a Paschal lamb is eaten by those who did not register for it. And if, on the other hand, the priest had not performed the ritual of the Paschal lamb on the first Pesaḥ, there is concern that perhaps this offering is a peace-offering, and it would turn out that this priest did not perform the ritual of the Paschal lamb at all.