Though he didn't think about the climatic effects, Arthur C. Clarke beat you to it by a couple of decades in The Deep Range. Not with sewer sludge, but he proposed fertilization and phytoplankton farming to feed the whales... (and there's more to it if you haven't ever read it. Though dated, it's still surprisingly prescient.
I'm not sure if the Pacific Gyre would work as well as the Southern Ocean. The Southern Ocean has plenty of nitrate and phosphate, but not enough iron, which is why iron fertilization works there. The Pacific Gyre doesn't have any upwelling, so it doesn't have nitrate, phosphate or iron (it's a long way from any continents in the mid-Pacific). Though sewer sludge would provide some nitrate and phosphate, and you could enrich it with a little bit of iron, the vast size of that oceanic region means that the scale of the amount you'd have to ship out there is really large.
I made the proposal specifically to sequester carbon in calcium carbonate as a sophomore taking college chemistry and oceanology (a great course).
It was an obvious thing to do.
Though sewer sludge would provide some nitrate and phosphate, and you could enrich it with a little bit of iron, the vast size of that oceanic region means that the scale of the amount you'd have to ship out there is really large.
We make a lot of sludge near coastal cities and are looking for better things to do with it. I'd bet there's a fair amount of iron in it as well as trace minerals that we probably don't know about. The transportation by water would be cheaper than we now use by land and in most cases the sludge could be piped instead of burning methane to dry it. I like the containment aspects of the gyre. The stuff goes straight down.