Many crop fertilisers have that issue, but fertiliser is not only one thing.

One hypothesis is that algae are limited by some single resource which could be gradually dropped out the back of cargo ships as they cross oceans, seeding carbon-absorbing algal blooms as they go.

(I heard about this nearly 20 years ago, so I assume that has either been tested or banned since then…)

I'm thinking mainly about the Haber-Bosch process, which runs on methane, and feeds more than half of the world's animal population.

As mentioned in a sibling comment, it could run on some other heat source, but that hasn't been demonstrated yet on the scale of agriculture.

Indeed, that's one of the main possibilities I guessed you were thinking of; for triggering algal blooms, one of the things suggested was iron. (And given it was immediately going into salt water, ore would presumably have been fine… if the hypothesis was correct).

Can’t fertilizer production emission be compartimentalized?

Fertilizer isn't a singular thing or a hivemind.

Terrestrial ag fertilizers don't have to be produced by fossil fuels.

The Haber–Bosch process used to make ammonium nitrate can be solar powered instead of using methane.

Phosphate rock processing could greenified for a green "green" phosphoric acid wet process.

Potassium from potash is mostly a passive solar evap process.

For seaweed aquaculture, iron sulfate is used because iron availability is the most frequent limiting factor. It's frequently a byproduct of steel production, in much need of greening itself, and produced in large quantities. Volcanic ash is another abundant source.

it depends on the final balance. As mentioned above the ratio for iron is 13k Carbon atoms sequestered for atom of iron.

Keep in mind the vast majority of agricultural yield growth the last century is giving these limiting nutrients as fertilizer for plants, so it is successful and worthwhile for land based farming atleast.