As you'll see, the first major engagement for the 31st Arkansas was Richmond, Kentucky, August 30, 1862 -- 1 killed, 9 wounded. This was followed by lots of marching, counter-marching and skirmishing. When the Battle of Perryville was fought, October 8, 1862, the 31st Arkansas was still 30 miles south, so missed out on that fight. They then returned to Tennessee in time for the Battle of Murfreesboro.
There is anecdotal evidence that the 31st Arkansas fought in the skirmishes around Farmington, Mississippi, May 8-9, 1862, but no official confirmation that I'm aware of. At any rate, the regiment left Mississippi in July for the Kentucky campaign, so missed out on the battle at Corinth.
I don't know of any numerical designation for McCray's Battalion; not does Bass mention anything about it.