I just setup my 'new to me' 40g breeder reef tank. I bought it from someone who had it running for 2 years so I took about 80% of the water with the liverock in icechests. Trying to keep the system as intact as possible. I completely underestimated how bad the fine sand would cloud up the water during the "rebuild" and cleaning, was worse than yours. Anyways, I ran my 4 powerheads, protein skimmer, and three biowheels with only the filter medium (carbon removed) and it was crystal clear in 24 hours. I had to rinse the filters about 6 times though as they were getting completely stuffed with dirt. The previous owner only had the skimmer, no other filtration than the live rock/corals but I left one of the medium size biowheel in place with only the cartridge (no carbon) for mechanical filtration and pulled the other two off as they were just hanging out on the front of the tank temporarily.
I'm not sure if removing the carbon helped or not, I just cut a slit at the top and dumped it out, I'm new to reef tanks and have two freshwater planted, from that I know with a "natural" setup you don't want the carbon so I carried that philosophy over with me. Either way, rambling now, that does sound like it's taking too long to clear up, do you have any other filters you could borrow temporarily if only to aid in the mechanical filtration? Possibly you could try a sponge over the intake to further aid in the cleaning. Could just be minute suspended particles that need to settle. If the cup trick posted above works maybe cutting off the powerheads/filter for a short period might help with the settling.