What a fascinating question!

It does indeed sound similar to a picador, but it wouldn't be right to call it that, firstly because we already have "picador" in your text and the two terms are being distinguished, but second, and more importantly, because the "garrochistas" of the pampas referred to here are not bullfighters but ranchers: cowboys, in effect.

The fact that "garrochista" means a pole-vaulter in American Spanish (which turns out to be irrelevant here, I believe) led me to wonder whether the gauchos practise bull-vaulting, which is extremely ancient and was practised in Spain at least until the early nineteenth century, as witness Goya's etching "Ligereza y atrevimiento de Juanito Apiñaniz en la de Madrid", also known as "El salto de la garrocha":
http://museogoya.ibercaja.es/obras/ligereza-y-atrevimiento-d...
Emulated by Picasso:
http://www.smb.museum/en/exhibitions/detail/pablo-picasso.ht...

I would imagine, in fact, that this may be how "garrochista" came to mean a pole vaulter. It's a tempting idea. But I think it's wrong here. I find no signs that bull vaulting is practised on the pampas; on the contrary, "garrochistas" do something else entirely.

What they are is already indicated in the DRAE definition: "el acoso y derribo, a caballo, de reses bravas y en faenas camperas de apartado y conducción de ganado vacuno". They are mounted cattle herders who carry and use a lance called a garrocha. The following is from a book on "The First American Cowboys" by one Donald Chávez y Gilbert:

"The 13th century knights and Spanish rancher / Caballeros developed a method of rounding up (rodear) and capturing cattle for branding, etc. borrowed from the knight's skill of jousting with a lance. This heritage of Knighthood was carried from Europe to the Americas in the 15th century. The technique evolved from the Caballeros use of the lance. In this case the lance is called a garrocha. It was a 12-foot long wooden pole with a blunt tip used by the Garrochista on horseback. The garrocha is carried and used in a fashion similar to the Caballero's lance. But instead of the Garrochista and the steer racing toward each other as in a knightly joust, the Garrochista chases after the steer. An Emparedor, a horseback assistant, rides alongside the steer to guide the steer toward the Garrochista. Emparedor is derived from the Spanish word meaning to hobble, or tie. The Garrochista lunges at the side of the rump of the steer with the blunt Garrocha and knocks the steer off its footing. The steer or other livestock tumbles, enabling the Emparedor to leap off his horse, bulldog and hold him down or tie the animal's legs."
http://www.nmhcpl.org/uploads/CH2.pdf

I think this is a term with such cultural specificity that the Spanish word should be retained, and preferably a helpful explanation added in English, as suggested above (there are possible variants).