I don't think the dishwasher will be "full of water" as it doesn't actually fill up - rather, it only uses 2 gallons maximum per cycle, about the amount that would be the bottom of basin of the washer.

That's what I meant. The water drawn from the dishwasher is small enough to not even purge the cold water from the line in many houses. So you would just be wasting heat by filling the pipe with hot water while only taking the cold water from it.