A digression on cost.
So, cost wise, it's more efficient to have a large army than a small one.
There's two kinds of cost - ongoing cost and capital cost.
Firstly, you have to ask whether this army is intended to fight wars by itself (as the prewar US Regulars pretty much were, possibly accompanied with some hastily raised volunteers to bulk out the force) or to serve as the nucleus of a larger army.
With the size of the ACW the latter seems obvious.
This means that either way you need the capital outlays associated with a large army - you can expect to be blockaded on the commencement of war, so you need the weapons in-hand as much as possible. So if your "mobilization strength" is 300K then it doesn't matter if your peacetime army is 50K or 5K - you need the weapons for at least a large fraction of 300K. Fortunately you'll probably have them at least until the 1870s, as Marsh/Snider/Tatabiere conversions for your rifle-musket buys are all a thing, but after that you'd better plan on making replacements (even if piecemeal).
So a small army doesn't save on small arms here.
Artillery, much the same situation with the problem that here you do need to buy modern - but this is the CSA and it's Tregedar who will be producing most of it, so a certain amount of buying will take place anyway because of the good old principle of pork.
This might mean CS Army peacetime arty formations actually do act as arty. They might well consist of 20 men and 30 *insert 1870s slang for black fellow* per gun, which would be a way to conserve on white manpower (and pay) somewhat without needing to arm any substantial numbers of black people. (Having this transition take place on mobilization would allow you to bulk out a peacetime establishment by a factor of 2 to 2.5 - which is useful.)
Cavalry they might skimp on a bit, cavalry militia is an almost inevitable result of the Confederate social structure (got to hunt down the escaped slaves) and historically "the dam-ned Virginia horsemen" scared the bejeezus out of the Union even before they'd done anything to earn a reputation - might be considered that they just need some scouts and let the mobilization on the event of war bulk it out to a useful scale.
A lot of the admin stuff (on the American scale of doing things) is sparse enough you don't save a huge amount by going for 20K instead of 50K (using those as baseline numbers). The Danish example above has only a couple of dozen men on the staff for example.
Engineers, well, the CSA will have more engineers than they need for strict war work because of the Mississippi. (A levee en masse, aha.) Again you don't save much by going small.
So what this means is that all the procurement is effectively a set cost. What's not a set cost is functionally troop salary.
Article on pay:
https://www.civilwarhome.com/Pay.html
Using the prewar values (where not provided multiplying Union officer pay by 0.92) and assuming a regimental structure as per 67th's post, an inf regiment would cost:
1 Col $195
1 Lt Col $166
1 Maj $155
10 Coys each of
1 Capt, $106
1 1Lt, $97
2 2Lts, $194
4 Sgts, $71
4 Cpls, $52
2 Mus $22
and 90 Ptes $990
Company total $1510
Regiment total about $16000/month
So a yearly budget increase of $1M would amount to about an extra five infantry regiments. This is where the budget can be adjusted up or down.
Given the value of cotton exports (cotton exports in the antebellum being about $250M per year
discounting what would now be export to the North but which was internal trade then) a few basis points on the export tarrif pays for an extra regiment of infantry. (A basis point is 1/100 of a percent.)
This would tend to mean that the army size question would be a matter of "what do we want" not "what can we pay for" - the marginal increase in the tax/tarrif burden for supporting an extra regiment is minimal.
Just don't find yourself stuck with an army of 30K troops with modern weapons in 1898 and a vast stockpile of old single shot breechloading rifles (and militia with smoothbore artillery) - Spain wasn't able to invade the US so it didn't matter but the Union will certainly be able to invade the South if you mess up like that!
Given the historical pattern of the second half of the 19th century, though, an important question is going to be how the South handles mobilization of reserves. Will they do conscription to impart basic skills like a continental European power, handle it with well-subscribed militia and volunteer associations like Britain, or "basically not bother" and rely on post-mobilization training from green recruits like the historical US?
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