Am new here, cannot think inside the box and have no chance of coloring inside the lines. I came here recently to be schooled, and the people who posted above are the ones have gen kind enough to teaching me.
Face with the problem of a half a trash can of "greens" sludge which probably has a glorious aroma, I would just start physically forcing dry "browns" into the trash can until mixture became semi-solid at minimum. I would probably go with a 50% of paper/cardboard and 50% high lignin stuff like leaves, bark mulch and woody stuff.
Once I got it somewhere around 2-4 browns to 1 Green (or dry enough to deal with), I would add it in as I turned my compost piles - several inches of active, then add a thing layer of the dried up sludge over half of the layer of active compost, then another layer of the active compost and another thin layer of dried up sludge.... goal with be to disperse it as evenly as I could through my active pile. Then turn again in a couple of days.
Paper has a huge amount of carbon, which is what you need, but in the end doesn't produce a lot of solid matter. Paper is mostly cellulose which breaks down into starches -> sugars -> simple sugars and is mostly consumed by the microbes.
The woody stuff, bark, etc. (high lignin content), takes a lot longer to break down and produces a lot more solid matter at the end of the process, which is what we want.
So, If I had a half a trash can of greens-sludge, I would be forcing ripped up newspaper into it feed the microbes and create heat, and forcing in an equal quantity of dry leave / bashed up bark mulch or other high lignin content stuff to get a decent amount of solid matter at the end.
May not be the best solution to the problem, but I would not have typed it out if I had any doubts about it working.
I don't believe we can resist the things which make no sense - I believe.