I don'have access to such riches, but it seems to me that seaweed would be a great compost ingredient. As mulch, wouldn't it become matted and stinky? Maybe the eelgrass would have a better structure.
I think I BUY seaweed (or at least kelp) dried and ground up and called "kelp meal" as fertilizer. Maybe that's your answer --dry and crumble on the driveway first. But I only use a handful at a time -- mulch maybe overkill.
Do you also have access to something called "salt marsh hay"? I've read about them as excellent weed-free hay, great for mulch. Combining the crumbled seaweed with other dry mulch matter before using seems like a good idea to me.