I did a search of Kansas regiments a few years ago, looking for any deserters from Arkansas regiments who may have switched sides, and I did find about 200 or so who I could identify as Arkansans. I haven't done that with Missouri regiments, as yet, due to the sheer number of Missouri Union regiments and the limited amount of time I can devote to it. I did do a search of one of the Union Missouri militia regiments rumored to have enlisted some Carroll County men and I found 20 or 30 of them -- can't recall the exact number off the top of my head.
I'm afraid the exact number of Arkansas men who served in the Civil War is never going to be known precisely. The best we can do is get as accurate an estimate as we can.
One of the estimates I've seen is based on adding the numbers of names on the NPS Civil War Soldiers and Sailors System. Brian, if you ever see that source cited don't believe it. The CWSS can't be used for numerical estimates for a couple of reasons -- first, the system lists every permutation of every man's name, and some men have records where their names are spelled a dozen different ways, resulting in that one soldier being counted 12 times; second, there are entire blocks of names missing from the CWSS -- for example, about half the Arkansas men whose surnames begin with the letter "K" are missing.
Researching Arkansas troops sure is fun, ain't it?