Well on MLE at least there is not 6 hours available as you also need to do breifings which should be completed before TOD. I would also like to draw the parallell to controlled rest where the resting pilot should not engage in any duties for 20 minutes after waking up.
Another thought, does the 3rd pilot need to be in the flight deck for takeoff and landing? OM-A says he should, if not resting. Since a First Class seat is reserved for rest, could the augmenting then be resting in the cabin during takeoff and landing?