I just played this scenario in Ultimate General: Civil War. It is, of course, the Antietam scenario.
I not only defended all my victory points, but actually counterattacked after the Union spent themselves on my defenses. I believe I eliminated more than 90% of their army.
In the real world, this would have been equivalent to the Union Gettysburg victory and would have marked the end of the war, with the Confederacy victorious.
In this situation, I expect that by the 1920s, the issue of slavery would have been on the ballot in the South, as machines were more economical than slaves.
Then, the Confederacy and the Union would have fought together in WWII against the Axis forces.
After that, maybe in the 1950's or 1960's, I could see the Confederacy joining the Union either loosely, or more formally, reuniting our countries... and this time, the concept of successful secession would have been proven as viable, so the North might have minded its p's and q's a little better this time.
It was an article of faith among these people, expressed to me more than once by my father and others, that the hand of God can be seen in the untimely death of Stonewall Jackson because it meant the ultimate loss by the Confederacy because he was not available to Lee at Gettysburg.
These Southerners saw God's hand in the affair to preserve the union to fight the second world war and to wage the Cold War.

I commend to your attention the 1961 (CW Centennial) book “If the South had won the Civil War” by MacKinlay Kantor, a noted CW historian and author. Your thoughts echo much of what he put into this book. One interesting variance was how the CSA conquered and incorporated Cuba as a state WHILE the Union failed to buy Alaska.
So there was a non-Castro Cuba but the Soviet Union had Alaska which was worse.