If you're trying to make it not-dark, why can't the horse just run the wrong direction? Say into the fire swamp
If you want the horse dead, go for it. A minor to minor-moderate wound would only add to the stress of the horse (and make it run faster to get away). A serious wound could ofc cause it to bleed out as it runs.
The horse could break it's leg right before it would drop from exhaustion anyways (clumsy, stumbles, snap).
Keep in mind most horses only run so far before thinking 'wait, I don't see that lion behind me any more, I guess I should slow down and see if it's kewl' so you need your horse to drop within about a mile or so.
And really, imo, not killing your horse but leaving it to die is MUCH darker than mercy killing. Even a knife to the artery, while not pretty (and a bit dangerous), is much better. Or at least try and go chicken shit and watch the horse die anyways if time allows. Just walking off is cold if he has time to find the horse to begin with.
If your MC doesn't try to help the animal stop suffering, or damn well try, you mega alienate me as someone who sympathizes with animals. Same if he left a baby with a fever to die with no wolves circling in (and even with wolves circling in, that's a nasty way for a baby to die, Classics be damned!)