My favorite home is net zero energy and I try to run it net negative for all inputs including food, trash, fuel, power, water, wastewater, and money. I have a dishwasher, but I've never used it and probably never will. As long as it remains unused, it registers zero for all inputs except trash for which it scores negative. Although it's in nearly new condition, it represents negative trash because it was salvaged from a high end remodel. My main reason for not using it at all is that it takes too much power, but if I did use it the power would be generated by the home and the waste water would go directly to the plants.
Likewise I never use paper plates for eating on. I do use them for compost and that helps me to maintain negative trash and food targets, but I never buy them or use them for eating. The main reason for this is that I am not able to calculate the impact. Even if I could work out the numbers it would be positive for water, energy and money. Paper takes a lot of water to produce, energy to transport, and money to purchase.
I realize that this answer doesn't fairly answer the question, but it explains the reasons for the strategy I've chosen and it certainly makes the calculations possible. If I absolutely had to choose between the two options you offered, I would run the dishwasher at a time when I had a lot of surplus power for the reasons stated above, but the equations would be much different for people who purchase dishwashers and haven't closed the loop for power, waste, money, water, and wastewater.
It might be interesting to note that I sometimes use banana leaves as a substitute for paper plates, and as compostable cookware that doesn't require washing at all, but I'm not sure if that pertains to the question well enough to be included here.