"Introducing animals to compete or kill native species, the same with plants, leads to situations with unintended consequences.

Best to not screw with that stuff."

Wiki:
"....
However, controversy remains over the effectiveness of atmospheric CO2 sequestration and ecological effects[2]. The most recent open ocean trial of ocean iron fertilization, dubbed LOHAFEX, was conducted from January to March 2009 in the Southern Ocean (Atlantic Sector).

Fertilization also occurs naturally when upwellings bring nutrient-rich deep-water up to the surface, as occurs when ocean currents meet an ocean bank or a sea mount. This form of fertilization produces the world's largest marine habitats. Fertilization can occur when weather carries soil long distances over the ocean, or iron-rich minerals are carried into the ocean by glaciers, rivers and icebergs."

Iron is very abundant on Earth and this universe- so the dust eruption from volcano will have significant amounts of iron in it, as will dumping sand or any kind of landfill or material from dredging operation.

"Dumping at sea is regulated under the Dumping at Sea Acts, 1996 and 2004. This Act implements the OSPAR Convention adopted in 1992 and entered into force in 1998 which provides for strict controls on dumping of material at sea.

The Dumping At Sea Act, 1996 (as amended) prohibits the dumping at sea from vessels, aircraft or offshore installations of a substance or material unless permitted by the Minister for Communications, Marine and Natural Resources."
http://www.agriculture.irlgov.ie/fisheries/coastalzonemanagement/dumpingofdredgespoilatsea/

So the first thing you do is get rid of this US law. Second you have the government actually provide a small amount of money for dredge material to be shipped far enough out into the oceans and to places which are thought best. Let's cap the total amount money per year at say 50 - 100 million dollar. So dredging can dump their material practically anywhere they want, but the govt will pay to have them ship it to locations that they think would provide the most benefit per ton dumped.
Considering that generally dredging would dump stuff out to sea near a river- that might be a pretty good location due to the river currents dispersing this stuff- not to mention that it's close to what nature does anyway when rivers seasonally flood.

You might want more effort in spreading this material around. But simply dumping this stuff in deep water should disperse it fairly well. Generally these guys aren't trying to muddy up the ocean, but if they knew this was desirable they could probably quite easily manage to do this fairly well. But dumping something a thousand miles out, is more costly than 20 or 30 miles out- so providing addition money could encourage those who able and it is practical, to do this.