Great idea, but not practical. It can be done, but the inputs will outweigh the outputs by at least 10 times. First, are you talking about kelp like what's in the ocean in the cooler areas in the United States? Or, are you talking about sea weed that you find in tropical waters?
If it's the coldwater variety you will need a chiller in an aquatic system to keep the water the right temperature. The bigger the system, the bigger the chiller. To keep a 275g system in the low sixty degree zone you're talking about chiller that's gonna cost you at least $1,000. That baby is gonna suck a lot of juice as well. Then add in the appropriate lighting...at least another $1,000 and the cost of replacing bulbs annually. LED would be an option, up front costs would be at least $2,000 but then bulbs would not need to be replaced for 5+ years. We haven't even scraped the top in terms of start-up costs yet either. Substrate, pumps, salt, test kits, water supplements and oh yeah...you're not going to be using well water on this system. You will need to be using RO/DI water.
If you're talking about the tropical variety, think red gracilaria variety, you're talking about better lighting, which means more heat during the warm months...thus the need for a chiller about 5-6 months of the year. More expensive than the cold water variety of kelp, but better nutritionally for us (red gracilaria is called "ogo" in Hawaii and eaten in sushi, poke and other dishes).
I've been into saltwater aquariums, and large systems, for almost 12 years now. I worked part time harvesting sea weeds when I lived in Hawaii. I love doing stuff myself and anything to do with farming, self efficiency, etc... I have built systems that did not use any form of mechanical filtration, instead using attached tanks (rubbermaid feed tanks) with deep sand beds, various sea weeds and live rock. For what it took to build these systems you will never get enough out of them to make it worth your while. Electricity requirements alone will kill you.
Chalk this one up to a "need" that you will have to trade for with the systems that you have that produce in excess for you.