"Is there a way to find what campaigns he fought?"
Your Question is talking about two different units at different places at the same time. So I'll try to answer and explain the part that effected Morgan Griffith as best I can.
As a Brigade Commissary Department Quartermaster he was basically a storekeeper for his Brigade, he may not have "fought" in any Campaigns. Don't get me wrong. Keeping up with the needed supplies of a Brigade of 2 to 3 regiments and over 1,000 men is no small, nor is it an unimportant, job. But that job may have been done far from any Battlefield.
Morgan Griffith enlisted in Co "I" as a part of the 19th Arkansas' reorganization efforts in the early spring of 1862 to strenghten this regiment after the losses that it suffered while in Oklahoma from illness. In May of 1862 Dawson's 19th Arkansas was ordered to Arkansas after Gen. Van Dorn abandon the state and moved his Arkansas army to Mississippi.
It was stationed at Camp White Sulphur/Camp Lee at Pine Bluff where it was brigaded with the 24th Arkansas and Crawford Battalion. It was again reorganized and remained there until September 1862 when the brigade was ordered to St. Charles and then to Arkansas Post, where a majority of the regiment was captured Jan. 11, 1863.
Those who were captured at Arkansas post were exchanged and consolidated into one Regiment the 19th/24th Consolidated Infantry Regiment under the Command of General Pat Cleburne. These men never returned to Arkansas until after the war.
Hardy's Regiment was a reconstructed "consolidation regiment" made up of the men of the 19th (Dawson's) the 24th Arkansas, and Crawford's Batt. Infantry that for one reason or another were NOT captured, or had excaped from Arkansas Post. Very few of the Infantry actually excaped from Arkansas Post.
Those men who became a part of Hardy's Regiment were those who where stationed at St Charles, and Pine Bluff, or were in the Hospitals at De Val's Bluff, Little Rock and Pine Bluff. None of these men ever when east of the Mississippi River.
Hardy's Regiment (actually's again Dawson's Regiment) was reorganized in February of 1863 from the men at Camp White Sulphur near Pine Bluff. It remained at White Sulphur Springs until late July 1863 training and as part of the lower Arkansas River Valley defenses.
It then moved with Col John Clark's Missouri Brigade of Frost's Missouri Division from Fort Pleasent to the Defences of Little Rock in August 1863. Where it remained until it was reorganized into Gen Tappen's Arkansas Brigade who it remained with until the end of the War.
Hardy's regiment most notable engagement was Jenkin's Ferry.
Please keep in mind that this is only a general outline.
By the way Morgan service record did not end in February 1864. Just the Confederate Armys ability to keep detailed records ended at that time. In most likelyhood Morgan served until the end of the war at one of the several Quartermaster depots in Arkansas and Texas.