This unit, as you know, began in the west serving there until late winter 1864 before moving east. Accordingly, they could have brought whatever western battle flag they carried. Based on their service and unit association I cannot pin down what pattern of battle flag they carried. They should have had a silk issue Polk Corps flag in April 1862 as they served in the district covered by Polk's corps. How long they carried this flag who knows.
As with the Tennessee regiments that came east in 1864 in Bushrod Johnson's Brigade who brought their blue and white Hardee pattern battle flags with them, losing two in June 1864 at Petersburg which were replaced with Richmond Depot ANV flags for the rest of the war. I use this as an example for what could have happened with the flags of the 1st Confederate Battalion.
As far as uniforms, they would be supplied by whatever depot was closest to where the battalion operated. The Confederate units had no special uniforms that I have ever heard of .