Years ago we used Pears Clear soap as it was/is biodegradeable, while bathing in the pristine Murrumbidgee. Compared with what comes down the river in a flood, some suds for 30secs every other day is not going to be an issue.
I do take the point about low lying, already damp camp grounds. No one likes camping in others run off. I make an effort to put the shower where the run off goes into the bush, away from the general area.