(a) The GILYON TOSFOS writes that the Gemara indeed could have given this answer, but it preferred to give a better answer.
1) What/how is the gemarra's answer better???
(c) The KOVETZ SHI'URIM suggests further that in the case of the Gemara in Bava Metzia, in which each of the two claimants presents Simanim to prove that the object is his, the two claimants have two proofs which contradict each other. To divide the object between the two of them would contradict both of the proofs. In contrast, in the case of the Mishnah here, neither claimant brings proof that the pigeon belongs to him. Therefore, to split the pigeon between the two of them would not contradict any proof to the contrary. (The clarifying factor of "Karov" does not apply here, since the pigeon is equally close to each of the coops.) (I. Alsheich)
2) The above paragraph appears in the insights to baba basra 23b; isn't the correct pshat that to divide the object would possibly contradict ONE of the proofs (not both-- as stated below) since one of the proofs (or neither) might be correct???
(a) The Gemara's answer is better because we do not have to suggest a difference between the cases that may or may not be the real cause of the difference in the Halachah, but we can rely on a known difference in Halachah between the two cases which would certainly cause the difference in Halachah. In other words the tree near a pit does not necessarily have to be cut down, so we cannot have it cut down in a case where there is doubt, but the tree near the city must be cut down in any event, so we may have it cut down even if there is doubt as to whether it pre-dates the city or not.
(c) We have two proofs. Each one says that the whole bird belongs to one, different, owner. To give each one a half would contradict both proofs, albeit each one only with respect to half of the bird. I would point out, however, that this aspect of the answer (that splitting the bird contradicts both proofs) is not found in the Kovetz Shiurim quoted.