The problem here is that the command situation in the Department of Arkansas was so very fluid and constantly in change. An order given one day would be retracted the next. Aug 1st, Hindman was about to be superceeded in his command by Holmes. Second, Hindman was expecting a large influx of reportedly 25 "new" piece of Artillery being brought from Mississippi by Gen'l Parson's Missouri Brigade when it returned to Arkansas.
So all of these orders to organize and reorganize these unit were all so short lived that you can only compare them to what organization that survived to October 1862 when the reorganization of the Confederate Army in the Department of Arkansas became more stable.
August 1862 was the time when the first unit Muster Rolls were being taken for the Units being raised. Some units that had been organized for as much as a year like Dawson's 19th Arkansas Inf. were almost completely reorganized and men being transferred to other units like Portlock's 24th and Hart's Battery.
The shuffling of new conscripts, men and companies from one Arkansas organization to another was so widespread that not being able to identify which Captain Brown (S.W. or Louis) was in command of any Battery does not surprise me. The fact that no Muster roll exist past October 1862 to help identify which Captain Brown we are looking for seems to indicate that neither actually came to command an Artillery Battery more than in a temporary position as a staff functionary of either Gen'l's Hindman or Holmes, who had given them a blank card to do whatever was necessary in getting these batteries off the ground.
This would seem to also apply to Lt. Henry Halliburton and Lt. William Leonard who were assigned to Captain Brown per orders. Finding out what happened to them would give us some indication about which Captain Brown. But this is the last we see of them in Official records as far as I am able to determine. Again what unit did they serve in after Oct. 1862? Their disappearence again would seem to indicate a staff duty maybe in the Ordance Department or just temporary detached duty.
After Van Dorn took everything not bolted to the floor, and even some things that were, from Arkansas to Mississippi, and until the arrival of the Artillery train from Mississippi with Gen'l Parson Brigade in August, where were all these guns coming from? We know that some were taken from Pike in the Indian territories. And some obselete guns that had been buried at the Little Rock Arsenal had been dug up. There could not have been that many unmaned artillery pieces just sitting around waiting for Artillery Companies to be organized.
There are many "Black Holes" in these records regarding Arkansas and you know this. We have pieced together the Infantry pretty well, but Artillery is another matter, For example according to Union reports of the Battle of Arkansas Post say the 17 guns were captured during that expedtion. 5 were from the CSS Ponchartrain, 6 were from Hart's Battery. That's 11 of the 17 reported. Who were the other 6 guns from?
Sorry I did not mean to write a book.