I'm surprised some of the more motorhead members haven't responded but I'll give you my take on it.
I don't think I would tie into the outlet side - might work but you'd have a pump pulling on a pump. If you tried it, I think it would absolutely require that the engine were on to use the washdown pump (independant of whether you wanted warm water) because I do not believe a washdown pump could pull through the intake pump when off - at best it would put excess strain on the washdown pump.
If I wanted the ability to have warm washdown from a closed cooling system, I think I'd put in a second exchanger and allow the heated antifreeze to heat the washdown water but that might cause issues too - for the motor or for the washdown pump.
Richard
Richard