The 22nd/35th Arkansas Infantry was organized at Camp Johnson, Arkansas, on July 11, 1862, so obviously this regiment was not present at Elkhorn Tavern.
The 22nd/20th Arkansas Infantry was organized at DeValls Bluff, Arkansas, on April 9, 1862, a month after Elkhorn Tavern. It is possible that a few companies of recruits intended for the 20th Arkansas accompanied the 19th Arkansas to Pea Ridge as reinforcements. This may account for the references to an authorized, but not yet organized, regiment being present at the battle.
The flag you described resides at the Old State House Museum at Little Rock, where it erroneously identified as belonging to the 22nd/20th Arkansas Infantry. In fact, it is associated with the 22nd/35th Arkansas Infantry, and is a postwar creation (perhaps for a regimental reunion). The battle honors displayed on the flag include two battles (Oak Hills and Elkhorn Tavern) which occurred before the regiment was organized, but which members of the regiment fought in while serving in other regiments. The remaining battle honors are for those engagements in which the 22nd/35th Arkansas participated in. In the past, I've provided documentation to the museum challenging the authenticity of the flag, but they still haven't changed the designation.
The 35th Arkansas was engaged at Prairie Grove in Fagan's Brigade, and was redesignated as the 22nd Arkansas during the general army reorganization following the battle. The earlier 22nd Arkansas, serving east of the Mississippi since April 1862, had been redesignated as the 20th Arkansas, because that number had been skipped by mistake.