What role did the Confederate regular army play in the war, and why did they have such small numbers?
Mostly officers. The senior Confederate generals held the regular commission as "full general." That included Lee, the two Johnston's, Bragg, Beauregard, etc. Also lots of staff officers, etc.
Although on paper some tens of thousands strong, the Confederate Regular Army only recruited a few hundred enlisted men (form memory), most of whom served artillery in the defense of New Orleans in 1861-62.
Nobody seemed interested in enlisting. Easier to get into a volunteer unit of the Provisional army. Only one year of service, elect your own officers, no humbugging about uniforms, etc. (of course that was in the first year, before the conscription).
Here's an account by Mr. Phillip B. Spence, who was commissioned in the CS Regular Army as a lieutenant...could get no recruits, and having no command, was placed on staff duty, including inspecting and mustering into service volunteer units.
There were some "Confederate States" units were formed but these were principally composed of volunteer companies assigned to them.
Confederate Regular Army post, CWT...
Confederate regulars, post, CWT