Tza'ar Ba'alei Chayim in Killing a Dying Animal

Is there a prohibition of Tza'ar Ba'alei Chayim in killing a dying animal?

The Gemara in Bava Batra (20a) discusses placing a dangling limb of an animal in a window so that it serves as a barrier against impurity, and the ruling is that it must be left there so that it is not removed, for if it is intended to be removed, it does not serve as a barrier. The Gemara asks: 'Cut it off and throw it to the dogs' (fearing that the limb will be cut to feed the dogs), and answers: 'Since there is Tza'ar Ba'alei Chayim - it is not done'. Tosafot (there, d.h. 'Since') question: 'Why not take the whole animal and throw it to the dogs', and similarly question the Rashba, Ramban, Rabbeinu Yonah in their commentaries, and the Ri Migash there, and from their questions it is clear that there is no prohibition of Tza'ar Ba'alei Chayim to give a whole animal to the dogs so that the dogs kill and eat it. Apparently, the reason is that the prohibition of Tza'ar Ba'alei Chayim exists only when the animal is alive and suffering, but not when it is being killed (even not through shechita and not for human needs). The Noda B'Yehuda (Tnina YD, siman 10, 13, and in Mahadura Kama YD, siman 83) proves from the Gemara in Chullin (7b), where it discusses harmful mules, that Rabbi said: 'I castrate them - there is Tza'ar Ba'alei Chayim. I kill them - there is Bal Tashchit'. From the fact that the Gemara does not mention a prohibition of Tza'ar Ba'alei Chayim in killing the mules, but only in castrating them, it proves that in their killing there is no such prohibition. In Tosafot Sanhedrin (80a, d.h. 'In the ox') it is written that it is better not to put an ox that needs to be killed into a cell, and rather it is better to kill it with a cleaver, and this implies that in killing there is not so much Tza'ar Ba'alei Chayim. Also in Avodah Zarah (13b) regarding 'making it a gistra' (cutting the animal in half) it is stated in the Gemara there not to do so with sacrificial animals because of 'do not do' or because it appears as if inflicting a blemish on sacrificial animals, but they do not worry about Tza'ar Ba'alei Chayim in killing. However, see in Sefer HaChinuch, mitzvah 452, and in Rambam, Moreh Nevuchim, part 3, chapter 48, where the reason for shechita is explained, and this requires study. Practically, it is clear that there is evidence that in killing there is no prohibition of Tza'ar Ba'alei Chayim, and although there are those who disagree, it seems more likely according to the law that as the sugyot imply, there is no prohibition. Therefore, it is preferable to kill a suffering animal immediately rather than let it die in prolonged suffering. And although in Sefer Kav HaYashar, chapter 87, it is brought from the Zohar (Yitro 68a) not to kill any creature without reason, and this is also known from the Arizal, however, here, since it is for the benefit of the animal so that it does not suffer, it is preferable to kill it.