Okay, I see what the problem was. The War Department messed up when it tried to verify Frank Elliott's claim of service in Brooks' Regiment.
The regiment commonly referred to Brooks' Regiment was the 34th Arkansas Infantry (Col. William H. Brooks). However, late in the war in 1864, Lieut. Col. Iverson Lea Brooks was promoted to colonel of the 26th Arkansas Infantry. If they had looked at Col. William H. Brooks' 34th Arkansas, instead of Col. Iverson L. Brooks' 26th Arkansas, they would have found Pvt. Frank Elliott serving honorably in Company G. I don't know why they failed to do so. The Official Records almost always refered to the 34th Arkansas as Brooks' Regiment, while the 26th Arkansas was only occasionally referred to as such, since Iverson L. Brooks was in command only late in the war.
Of course, that still wouldn't have solved the fact that Zimri Franklin Elliott doesn't appear as such on the regimental rolls. If your g-g-grandfather had been able to obtain affidavits from former comrades or commanding officer that Zimri Franklin Elliott had served as Frank Elliott, he would have qualified for the pension. However, as he was living in Texas when he applied, I imagine that he had lost contact with his old Arkansas comrades.