For several years, I’ve worked on design teams for political-military simulations that include alternate histories of the Civil War.
Generally speaking, the design teams have always agreed that the die is already cast by 1860 for reasons of economy and political economy: if the United States is in the hands of a skilled played, the Confederate States can have no chance of winning a general war. The idea is that they will simply be out-produced and out-governed by a much more populous, much richer, much more industrial, much more cohesive northern neighbor. We therefore tend to design the alternate history for North America with the intention of providing the South with as much an advantage as possible. Since your intent is to develop a fun wargame that leaves some possibility for a Southern victory, you may find my comments interesting. (I certainly hope that you do!)
We begin, like Turtledove, with a Southern victory at Antietam. We then followed that with Anglo-French intervention by land and sea. As in Turtledove’s narrative, the South manages to leave the Union with Kentucky, portions of what is today West Virginia, the Unorganized (later, Indian) Territory, and the Confederate Arizona Territory (south of the 34th parallel). Significant populations of southern sympathizers are presumed to remain in Maryland, Delaware, and Missouri.
Like Turtledove, we also posit that a second war sometime in the early 1880s legitimized Confederate annexation of Chihuahua and Sonora. We judged that the Confederate Army would have benefited from a more pronounced martial culture, whereas institutional military training would have languished in the North owing to the bitterness felt over the resignation from the regular army of so many West Point graduates in 1861. As a consequence of obtaining British and French intercession, the Confederacy was forced to effect manumission, although it is quickly replaced by a system of what would today be called petty apartheid and debt peonage. (In other the words, Southern society, and the black predicament, are essentially unchanged.)
Before, during, and after the Second War Between the States, the Confederacy builds substantial fixed defenses along the Ohio and Potomac Rivers. During the war, they effectively employ cavalry (and a small camelry) in the West, along with commerce raiders that prey on Northern whaling fleets.
By 1914, the Confederacy has industrialized, but only to an extent. It is held back by (A) the weakness and poverty of its federal government, (B) the high number of uneducated persons relative to the total Southern population, and © studied attempts by both the British and French to keep the Confederacy from emerging as an independent competitor, as well as to prevent intrusion of Confederate raw goods into markets reserved for trade from imperial colonies. Nonetheless, the Confederacy has a small arms industry centered around the Tredegar Iron Works at Richmond and the industrial city of Birmingham, Alabama. Additional (minor) centers of industry have been established in Charleston, Savannah, and New Orleans. There is mining in Confederate Arizona, oil in Confederate Texas, and helium in Confederate Arkansas. Like the U.S., the Confederacy would have built a trans-continental railroad by the turn of the century, linking Richmond and Guaymas. It is up to you whether the Confederacy would have invaded Cuba. There was certainly precedent in the form of the filibusters. For the sake of bolstering the fortunes of an imperial Castille, I did not give Cuba to the Confederacy in my game design, instead awarding Richmond a client regime in the form of a Nicaragua overrun by William Walker’s filibusters (the point-of-diverge that I used was well before the Civil War; as a point of interest, we retained both Byzantium and the Crusader States in the Near East). You might give Cuba to the South to enhance their economy, although left in Spanish (or even placed in British) hands, it could become a convenient crossroads for contraband.
Probably both the Union and Confederacy sponsor Indian raids across their mutual border. One imagines that cavalry, camelry, armored cars, armored trains, airships, and aeroplanes will be the preferred weapons of war in the West, with the occasional armored car making an appearance. In the East, expect to see trench warfare.
Fighting a mostly defensive war, the South would have an early advantage – important given a relative dearth of heavy artillery. Richmond would also expect the British to substantially reinforce the Canadian garrison as a counterweight to Northern might. Probably the Great Lakes would be teeming with pocket battleships specially designed for those waters.
The South might be expected to be a pioneer in airship and aeroplane use given its size, including, perhaps, the first to attempt to launch planes from a converted merchantman or cruiser. I imagine that they would have borrowed from the French concept of the merchant cruiser. Probably, given the cost of heavier battlewagons, their few capital ships would have be no better than battlecruisers.