The U.S. didn't care about any of that when they did it in Canada, and the world didn't seem to care that much.
And I'm not saying the Americans are going to start atresting anybody and everybody off the street. I mean, they can do it and will to get people associated with the Freedom Party or who are shelling out anti-union propaganda. The Southern people won't mind as long as the Union only goes after them and leaves their everyday lives more or less alone, as was the case in the TTL occupation of Germany. We have to consider that the Freedom Party were probably already drawing the ire of the southern people even before the revelation of the holocaust, much like it's TTL analogue. A lot of the Germans had grown bitter and disillusioned with the Nazis as they were perceived as needlessly continuing a pointless war and destroying the German state. It is most likely the same here. It was quite clear that the Soutn was losing and badly, and that Freedom Party was incapable of winning.
With a discredited Freedom Party, which was being purged, there is no possibility of an insurgency. The remnants Freedom Party were the only people who could reasonably challenge American power, and the population would most likely not trust them and see any point.
This doesn't even cover the fact that insurgents would need to get a lot of military supplies from a big source and be able to move them around in mass without drawing the attention of a most-likely counter-insurgency trained, professional U.S. Army, which would be practically impossible. The U.S. controls every major land route and waterway to move them around, and they would most likely secured or destroyed every major arms depot and stash. Military police rule the districts and local police would most certainly be run by sympathizers. And even if they could do it, no country would be willing or able to assist them or supply them.
Again, not saying an insurgency or rebellion is impossible, but chances are heavy against them succeeding. Most insurgencies and rebellions fail more often than succeed because established power always has the upper hand. More often than not, rebel victories come down to the establishment making a critical mistake. The Iraqi insurgency only happened at all because the US failed to secure many arms depots and money caches, disbanded the Iraqi Army and police without pay, failed to keep the infrastructure running. Even taking advantages of these errors, the Iraqi insurgency was defeated. I just can't visualize the efficient, experienced, and professional US Army of TTL making such rookie errors.