Was there any way of reconciling the Republicans' desire for equality for the ex-slaves with the ex-Confederate desire for self-rule in the South?
While it is impossible to know what might have been possible in any given situation, I would argue that there was no real way to reconcile these two positions.
The entire society and economy of the South before the war was based on the idea of white supremacy. Given this fact, any Southern "self-rule" was necessarily going to involve a return to white supremacy. It seems like it would have been essentially impossible to have self-rule with black equality.
The only way the two could have been reconciled would have been for the white Southerners to have a change of heart. They would have to have accepted the idea of giving African Americans legal equality. If they had thought about it, it shouldn't have been such a big deal since whites were still going to have the majority of the economic and political power in essentially every community. Absent such a change of heart, it would have been impossible to reconcile the two.