Where I live the lake fish are all on birth control due to the ladies at the beach relieving themselves in the water. There is a number of pollutants in that water, and people often get sick after swimming, ear infections are common. Some of the residue will not disappear on dried dishes, it will just be dry and you will eat it. gross.
There are filters that will clean it, but they are slow-drip and will be expensive for washing dishes.
20 feet elevation gain will be slow but doable with the right pump. The distance is not going to matter as much.
If this is for a semi-permanent location, or your own property then you can make a filter with a 55 gallon drum, sand, rocks and other things. Do an internet search and it will be easy to find. This could provide water at a fast enough rate. I would spare your RV tank and get an external holding tank if possible, but thats up to u.