I would like to discuss some aspects of צבע"ח. This topic is very loaded emotionally, and it is crucial to neutralize one's emotions while discussing it Halachicly.
- There's no Halachic reference to animals other than being a property. As such, one is free to make any use of it including making it suffer or killing it (don't rush!) as long as one benefits from it. If one does not benefit from injuring or killing HIS animals he's probably transgressing בל תשחית, but nothing from יורה דעה. (See JT Shabbat 7,2 explanation of עורות אילים מרודמים that the animals were beaten to make their skin red, or Soyto 9-10 about letting the blood of an animal before sacrifice). This approach is called scientifically "Anthropocentric Ecology" or Anthropocentrism.
This is פשיטא from the explicit verses of the Torah:
"ויברך אותם אלוקים ויאמר להם אלוקים פרו ורבו ומלאו את הארץ וכיבשוה ורדו בדגת הים ובעוף השמים ובכל חיה הרומשת על הארץ"
"ומוראכם וחתכם יהיה על כל חית הארץ, ועל כל עוף השמים, בכל אשר תרמוש האדמה ובכל דגי הים, בידכם נתנו. כל רמש אשר הוא חי לכם יהיה לאוכלה כירק עשב נתתי לכם את הכל"
The same logic applies to Hafker animals, that one is free to catch or make use of or kill for any need incl. fun (Nodah BeYehuda openly permitted hunting for fun only, but many opposed). Therefore, killing an animal is defaultively allowed and by itself does not fall under צעב"ח at all.
The whole concept of צעב"ח is questionable as all the Mitzvot we learn it from have some weird limitations, e.g שילוח הקן is limited to עוף טהור, אותו ואת בנו is limited to defectless animals, פריקה וטעינה is limited to חברך etc.
Therefore צעב"ח was never unified in any form of Halacha, and it is only mentioned "by the way" when other Mitzvot are discussed (Se Rambam Avida, Rotzeah also in Shu"A)
Most Poskim agree that the concept is used not as a practical Mitzvah, but as a tool to exercise one's Midos in order to mimic Hashem's qualities ( חסדים ורחמים). Therefore, the Mitzvah will not be the חפצא - to help the animal, but גברא - to use the situation to exhibit and exercise good Midos. The story about Rebbi (BM 85a shows exactly that - he did not show his sympathy), or the opposite - to prevent developing bad Middos such as cruelty (See Rem"O אה"ע סי' ה' סעי' י"ד).
As the matter of Midos, two people might be acting completely differently and still "performing the Mitzvah of צעב"ח" - One might express his compassion by killing it immediately, another might only pet it, and still another by spending a fortune to find a cure.
I'm sorry to stop, I can go on and on but hope this will suffice.