On the Sabbath..

See Shemirat Shabbat Kehilchata (27:57)..

Assuming the horse is in significant pain, the rule is that a rabbinic prohibition was permitted because of suffering of animals (Babylonian Talmud, Shabbat 128b).
As for injecting an animal on the Sabbath, it says there in Shemirat Shabbat Kehilchata (ch. 35) that it is permitted to inject into the muscle, since in that case there is only a prohibition because of grinding medicinal ingredients, which is a rabbinic prohibition. But if one injects into a vein, where there is then a Torah prohibition because of threshing, it is forbidden. See there the sources he brings from Pri Megadim and the Mishnah Berurah.
However, if there is danger that the horse may die, then there is room to permit it also because of substantial financial loss to the owner, even if the horse is not suffering very much. And here too, simply speaking, what is permitted is only a rabbinic prohibition (and even that is not simple).
Seemingly, one could make this depend on the dispute whether suffering of animals is Torah-level or rabbinic. But straightforwardly, the dispute is about causing suffering to animals, not about the obligation to prevent suffering that it already has anyway. True, some linked this to the obligation of unloading and loading, and seemingly the obligation of unloading is because of suffering of animals, and that is preventing suffering that it already has (certainly if the one unloading is someone else and not the owner who loaded it).
And if the animal is suffering greatly, it seems to me that one can permit it in ways involving an unusual manner of performance, or where two people do it together, even for Torah prohibitions.