As long as the distinction is drawn that there is a big gulf between the Average Joe Southerner and the CSA elites that they put to trial and hang, I think it could go off without too much grief.
On the plus side, it may prevent what in OTL proved to be the saving grace of the planter elite and the doom of Southern blacks - that the previously tense relationship between the poor whites and the gentry reconciled. Here, the USA would preempt that, play off of that tension, and could well prevent it from every happening.
Ironically enough, by using one of the OTL Lost Causer's main tenants - "you folks were just fighting to defend your homes, you never would have had to if those bastards in Richmond hadn't forced you to fight their rebellion." Only this time, it would be used to see the proto-Lost Causers hung. Ain't AH grand