You did say earlier that you cook once a day, meal for two. When I do that, all dishes for the day take a third of the maximum load on my machine, so I could wash once every three days, therefore averaging like 3 L per day tops? You handwashing every day are spending 6-8L daily which is more than double.
If it is true that you can spend <8L for an arbitrarily large amount of dishes, though, then I guess there must be an amount of dishes that you will outperform a dishwasher. They cannot handle an infinite amount of dirt, unfortunately. If you hand wash every 7 days you will be averaging less than 1L a day which really does sound unbeatable.