I can understand your concern for the lake that you sail on. It would behoove you to use a dishwashing detergent and or soap that will be kind to the lake water.
Where I live on a lake, the fishing is great but nobody in their right mind would want to eat any fish that came out of there because they glow in the dark. 
However the tidal river that flows through my city has cleaned up through the years in spite of the fact that most of the rivers and brooks flow into it. In fact, it's the same river where I keep my boat moored about 7 miles from where I live.
Just two weeks ago we had a lightning storm that hit some sewer pump motors in Fall River Ma., and the sewage overflowed right into the Taunton River. It took about a week to clear out.
So you see, I really don't share the same concerns that you do because there are worse things going into my river and the bays than my dish washing detergent.
When I wash dishes on my boat I usually heat up a tea pot of water and use Dawn Liquid Detergent with a scrubby in the galley sink. I set the soapy dishes out in the cockpit to rinse later.
For rinsing, I use either a clean one gallon weed sprayer filled with water, or my Stern's Sun Shower which I now store on top of my sliding companionway hatch. I don't even need to set the shower on the boom for this. It will rinse off large grills or pots right where it is.
I cut the wan down on the weed sprayer and made it a little shorter. Penelope and I use the weed sprayer for it's cooling mist on an extremely hot day and it also serves as a good way to rinse off dishes. If I have a messy grill or pot to wash, I will rinse it out overboard before washing it in the galley sink. The galley sink can get a little messy. I put all my dishes in this cute little dish strainer and mat that I have on board and they dry in no time.
What I would do on your boat is use a bucket with a rope tied to the handle and fill it with lake water. Then use this water to clean the loose food off your dishes first before washing them in the sink. You could rinse them in the sink or out in the cockpit with clean water. My boat has less freeboard than than most of the larger O'Days which allows me to just lean over the side and gut a large fish or clean loose food off a plate or pan.
I take all my showers right in the cockpit with the bag on the boom. The water drains right out through the self bailing system.
Another thing that I do is use Dawn to lubricate my sail slides. A couple of drops of Dawn on each sail slide is all you need to get them sliding beautiful. The rain will wash out my mast track nice and clean later on.
