Update: I took the opportunity to drain the pond, and vacuum out the sludge using my shopvac (what an amazing tool!) There was a solid inch of muck in the bottom that I removed.
Refilled the pond with my fancy water meter, and found that my pond is ~1600 gallons, and my bog holds an additional 250 gallons. I am shocked by how much volume the gravel takes up. My calculations estimated the bog at around 1000 gallons, and I assumed half of that would be take up by rock, but 75%??? Pretty wild.
The pump is running, and the initial water from the bog was a bit cloudy, but it seems to be running clear now. The ducks are doing their best to change the clarity of the water, but for now... it holds.