Hmmm, I think the main reason for "pro-Davis" and "anti-Davis" factions was mainly because the C.S.A. was facing an existential crisis at its birth on the losing side with many former U.S. politicians being at its helm. That being said I'd bet those two factions would evaporate in the face of Confederate victory in the War.
As far as political parties forming in an independent C.S.A. I'm not sure it would have happened, and that if it did, it wouldn't be until say WW1. If we look at how political parties formed in the early days of the good ole U.S.A. the forming of them was seemingly driven by cultural differences more than anything else. The Federalists were strong in the North mainly around urban areas, and a lot of rural ones, and Democratic-Republicans of Thomas Jefferson in the South. The Federalists were driven by the concept, (and I'm oversimplifying all this), by the principle of a strong central government to strengthen the U.S. to survive, and the Democratic-Republicans were formed out of a strong belief the Federalists would make the U.S. an over-bearing monarchy, (not sure that was justified), and were champions of limited government, with powerful State's Rights over the Federal Government being one of the main planks.
Looking back at how the fear of reverting back to what had existed before, the cultural and economic distrust between North and South, and the same distrust between rural and urban areas as well, one can say a similar but not identical situation could have existed in an independent C.S.A., but the C.S.A. would be starting out with a government unlike the U.S. one, being more what can be described as in-between the U.S. Articles of Confederation and U.S. Constitution in power, rather than a carbon copy of either. Before the U.S. Constitution was signed there were no political parties, the central government wasn't powerful enough to warrant their formation, BUT there were factions, usually political alliances between State governments against the others. I think something similar would have happened in the C.S.A.
In the hypothetically independent C.S.A. I don' think the central government would have been powerful enough for some time to bring about the catastrophe of political parties upon it, but they probably would have started to form in the aftermath of the WW1 era, as I think the times would driven many to the belief that a stronger central government for the C.S.A. was needed. Before that one would definitely political factions, my bet is that they would change frequently, but I think they would have mainly been sticking together and their reasons simply changing. My bet on factions would be:
Faction 1: Virginia, North Carolina, and if it was a member of the C.S.A., Kentucky
Faction 2: South Carolina
Faction 3: Georgia, Florida, Alabama
Faction 4: Tennessee, Mississippi, Arkansas, Louisiana, and Texas
And if the C.S.A. "won" it, the Indian Nations probably would have aligned with Faction 4, but been more of an "associate" as I doubt they would be a State, or States.
As for how I came to that conclusion of those States being in such factions with each other, its cultural, and economic, those two factors would have been the catalyst for it, similar to the U.S. and the States mentioned would have tied to one another by those things through roughly similar culture, and economic ties probably dependent on railroads and ports for goods more than anything, and they probably would not get along with other "factions".
As for their ability to work together, I think Factions 1 and 2 would stick together, but not be the same for, reasons, and I personally think they would have instigated a Confederate Civil War. By that I mean them trying to gain a firm hand over the other States/Factions in way of them being the center of power in a more powerful C.S. central government, and I think that would have happened before the end of the 1870's after Southern cotton lost it economic dominance, which was inevitable. In a Confederate Civil War I think Factions 3 and 4 would have stuck together and I think they would have won, after that I think steps would have been taken to make sure it never happened again, probably through a Constitutional Amendment or two, (and I also suspect slavery would have issue in the affair, and would probably come to an end afterwards as I think it would be more of a hindrance to the C.S.A. by that time), and after a terrible war for independence in the 1860's and a terrible civil war in the late 1870's the C.S.A. would probably be in bad shape.
After that I suspect the issue of re-unification with the U.S. could come up as an issue between the Factions, but whether or not it would succeed would depend on the actions of the U.S. during a Confederate Civil War. If they sat back, and did nothing, re-unification would be possible, if the U.S. backed a side, it would depend on which side they backed, and if was winner or loser. But if the U.S. worked behind the scenes to encourage the civil war, and took advantage of it to "occupy" areas of the C.S.A. or try a full on invasion, it would have united both side together against the U.S. and made the U.S.A. and C.S.A very bitter enemies from then on, with re-unification never being an issue for the political factions of the C.S.A.
After such happenings the political factions of the C.S.A. would have probably would have constantly bickered with one another, and rattle sabers when ever an important issue came up, mainly because the economy of the C.S.A. would be in a major slump after two factitious wars in less than twenty years. But that would probably change big time in the late 1890's and early 1900's after a certain oily discovery in Texas, and the discovery of the resources availability in other areas of the Deep South, and seeing how Factions 3 and 4 would already be the political powerhouse of the C.S.A. after a hypothetical Confederate Civil War, I imagine peace and harmony for all factions would follow, as disagreements tend to come from stagnation, and harmony comes from mutual benefit. After the discovery of oil the States of Faction 1 and 2 and some of Faction 3, they probably would have turned to small quality industrialization as their economic "cash crop" and together with oil in other areas would have led to some harmony as while different, the various cultures of the South are similar, and when there's mutual benefit the need for political factions, or full on divisive political parties would have been unnecessary. Now when WW1 comes along I've no idea on what would have happened there, and I think any hypothetical scenario on the subject can be accurate in any degree, as the variables after when the world went mad are an unknown.
Sorry my hypothesis is so long, hope y'all make it through all of it.