Based on the information that I have, I can not find any argument with your interpretations.

I feel sure that Hindman's mention of five companies in Chew's battlion was simply a mistake, since he had previously stated that the battalion was made of four companies.

Dec 14 ............
"Chew's Battalion consists of four companies and had been selected as a Battalion of sharpshooters. I assigned Major Chew to the command of it sometime ago. ........

Dec 23
Hindman to Gen Holmes-
"In Fagan’s command there is a battalion of five companies formerly commanded by Major Chew, which I organized as a Battalion of sharp shooters. It acted with and as a part of Hawthorns Regiment in the recent battle and Chew was killed,
................

Perhaps another question would be:
What became of the other companies of Gibson's battalion of five companies of Mounted rifles after some were dismounted to ultimately form Chew's infantry battalion? Perhaps transferred to Carroll's own cavalry regiment?

Alexander Papers [excerpted]

July 27, 1862
Fort Smith-Col Carroll to Hindman
Major Gibson's command is mustered into the service... at last report... five companies total strength 375 men with only 170 guns

August 28, 1862
Madison county Arkansas
Major cmdg. First battalion of five companies
"some absent with permission of Col Carroll"-Major William Gibson