Thanks for the suggestion on Human Thistlethwaite!
I have had fun figuring that one out over the last few days.
Also, I didn't really like how he breaks G2 when forming the corners into G3, so I looked for a different way to do that. So far, I think it's easiest to see the corner situation by just solving them
The bottom layer corners are pretty easy to do, and then in the top layer, a 3-cycle that uses only G2 moves is:
(F2 D R2 D' F2) y' (F2 D' L2 D F2)
It is pretty intuitive. The first 5 moves are simply a quick way to swap two corners in the U layer. The side effect is that two corners on the FL edge get swapped too. The second half is just the mirror of the 5-move sequence, applied so that it cancels out the side-effect. The result is a 3-cycle.
Without the cube rotation, the sequence is: (F2 D R2 D' F2) (L2 D' B2 D L2)