Yes, I imagine such a civ would just scoop up hydrogen from the Sun and run something like an artificial CNO cycle in reactors with that fuel. It makes more sense than using a dyson swarm.

What I meant there is that you can out-power the Sun's output easily even without that, by using the fuel on outer gas giants too. So technically you do not need to do anything with the Sun in order to "reach Type II" (it's defined by power generation, regardless of source).