You can wash and rinse a full load of dishes by hand with 3 gallons of water? Seems possible only if you aren't rinsing, but a dishwasher rinses too...
Do you own very few dishes or something? That's the only reason I can imagine not being able to fill it before needing to run it. Even before having a kid my wife and I could easily fill the dishwasher daily since we cook quite a bit.