"Chopo", by N. Howard "Jack" Thorp.

"Patonio/Platonio/Plantonia (etc.), Pride of the Plains"

"Doney Gal"

"There's a Bridle Hanging on the Wall" - Carson J. Robison.

There's also the poem "Ladrone", by Hamlin Garland, that you might trim and fashion into a song.

Cowboys (particularly in the days of the trail drive) would not ride just one horse, but several, selected from the remuda. Their relation to their horses was one of work and shared travails--as they'd speak of fellow hands--more than of deep personal affection. So authentic songs about the connection between cowboys and horses mostly reflect the work, and the sentimentalism or esteem is phrased in that context, as the songs I've listed demonstrate.