I never rinse seaweed in my high rainfall climate.
From what I understand, most sea salts are very valuable soil nutrients
and sea water contains pretty low levels of sodium chloride (aka the salt you
don't want in the garden)
I would use basically
any sea stuff I could drag home
I know people over here who burn anything 'shell' after eating the insides,
and add the high-calcium ash/char to the garden.
Think mollusks, bivalves, crustaceans, sea eggs...
The main thing I'm wary about is the potential smell-
if the system's not set up right, you can create a mighty stench, and I live in suburbia.