But having real trees above the water is much more useful than simple carbon capture, they would serve as wavebreakers for seasteading settlements too.
There is also emerging technology for nitrogen fixing that could work - Electrocatalytic Nitrogen Reduction.
All this being said, if the goal is carbon sequestration building a viable business is going to be next to impossible.
The easiest way would be to use your renewable energy to pull bicarbonate out of the ocean directly with electrolysis, reverse-osmosis, whatever magic tech the wizards in MIT are cooking up.
Only then you've got the problem of what to do with it all once you've got it on hand.
To bring nutrient rich water up and to generate energy in the process temperature difference between surface and deep water can be used [1]. So if this kind of tree is created, living on open sea can become cheaper and more pleasant than even living on a beach.
[1] https://en.wikipedia.org/wiki/Ocean_thermal_energy_conversio...
Ocean thermal energy conversion - I hadn't heard of this, this is awesome.