How about like a medical pump? Have a flexible polyurethane tube that's squidged by cams to pump the seawater through it. The cams stay in the dry.
Whenever we try playing god it seems we generally mess things up a few times so we'd need to monitor the actual outcomes in reduced CO2 in the atmosphere over several decades. It seems extremely high risk to assume there won't be unintended consequences...
Sequestering algae is how petroleum is produced. Does petroleum contain only hydrogen and carbon?
https://en.wikipedia.org/wiki/Azolla_event
Where "event" in this case is something that happened over an 800 000 years interval.
Time to buy a condo in the Greater McMurdo area?
"phytoplankton are estimated to capture 10-20 billion tons of carbon dioxide each year."