Don't kid yourself - you won't be able to down 2 cups of oil. No way. More is not better.
My optimal amount is between 2/3 and 3/4 of a cup. I would rather flush with only 1/2 cup than a full cup which is too much. More oil doesn't produce a better flush. Often worse.
I think its good that you eat a lot of fat. Cutting it out will mean your body will store a lot of bile in preparation for the flush, which should help.