I think in most cases, the C.S.N. would have acquired their small arms from the Army, or rather whatever the Army couldn't, or didn't use. For example, and I'm working from memory, I believe when the C.S.S. Atlanta was captured there was something like forty Maynard Carbines on board, which I would think they had them because the Army didn't have enough ammo for them, again working from memory. The C.S.N. was gonna be on the bottom of the ladder on small arms, the Army needed those more desperately, an example of that would be at the Battle of St. Charles in Arkansas, the C.S.N. had a vessel there, (the name escapes me), and they sunk their ship and others to block the Union Navy, and besides the cannons from their ship the sailors were armed with "single shot pistols" their only small-arms and had from their ships-stores and had to fight off Union Infantry with them, (didn't go very well).
When researching C.S.N. small arms you just have to go with records and first-hand accounts, and they were NOT given sporting rifles, ("Hawkens") , and would get standard Infantry weapons, (I recall Springfield rifle-muskets, 3-banders, were taken from the C.S.S. Tennessee), and that isn't different from the U.S.N. who never had enough of their Naval pattern arms and were more than likely gonna have standard Army pattern weapons in their ship's stores probably 8 times out of 10. I don't see any reason to assume it any different with the Confederate Navy, and one important thing to remember is small-arms weren't an issue item for every sailor, small-arms were the property of a ship, and were issued out to whomever as needed, with the basic issuing being, rifles and or muskets with a cartridge box, belt and cap pouch, (all kept with each individual gun), issued out to each individual in a landing party, with maybe one or two sidearms issued to the men in the boat/landing craft and the items returned to ship's stores once back aboard.
(I say the above procedure because too many Naval Reenactors, in fact all I've seen turn out with enough guns per man to make the Dismounted Cav. blush, authenticity in those two areas are severely lacking)
Of course what I've mentioned doesn't include C.S.N. commerce raiders from Europe, their ship's small arms weren't going to get anything from this side of the Atlantic, usually, and could pick up Enfields, or in the case of the C.S.S. Alabama a shotgun or two from where they originated.
Before I forget I just recalled an account from the C.S.S. Sumter where their small arms were mentioned to be double barrel pistols brought aboard before they left New Orleans, now I'm gonna have to hunt up where I read that.