My understanding as to why he sent Gregor rather than a more... professional lord, Is because Gregor is somewhat expendable. His family was only recently raised to nobility, and the Cleganes have few to no powerful allies to avenge their lord. Also, Gregor is known to be somewhat of a wild card.

The reason this is important is for the sake of justification. It's not a secret that what Gregor was doing was a straight up war crime. If any lord were bold enough to call Tywin out for allowing such crimes to be committed, Tywin could easily have made a public statement staying Gregor was acting on his own accord and turned rabid. He could then execute Gregor as "justice" for his crimes. With no powerful allies to seek revenge against Tywin, the Cleganes would just have to swallow their tongues and let the incident go. That is, unless, they wanted to be the next Reynes.

Had Tywin just decided to publicly confess that Gregor's actions were done under orders, those lords bold enough could use this as a way to rally support from the rest of the kingdoms for a crusade against the Lanisters to bring Tywin to justice, similar to Robert's Rebellion.