Just reiterrating a lot of what Seasick mentioned... it would be easy enough to T into the shower plumbing (after the pump, but before the shower fitting). What size is the tank? You may go through that very quickly - check the specs on the pump for the fow rate and do the math. Another issue could be that it doesn't flow enough to support running the engine - so in that case, just use the hose adapter on the engine. If you go this route... do you use the shower at all? If not, I'd probably mix the SaltAway (or whatever you decide to use) right into the tank.
I assume you have at least 2 engines, so obviously whatver size tank you have gets divided into two. I suppose you could get a couple jerry cans and fill them at the dock before going back to the mooring.
Or... here's a crazy thought... do your flushing at the dock and then tow the boat back to the mooring with your dinghy.
Do you have room for a larger/second tank?
EDIT: You could also fill a fish box (after cleaning it - don't want big scales going into the engine) with water and then use a little pump to transfer it to the fresh water tank. Or, just use the portable pump to push it right into the engine. A small bilge pump would do it - put some aligator clips on the leads so you can just hook it direct to a battery.