Shawn: Sounds like you've gotten quite a nice living space!
The 6" round duct may be a challenge. Might try this: A 6" duct has a cross sectional area of about 28 1/4 square inches. A 3" X 10" rectangular duct has 30 sq inches. You might try putting a round to rectangular converter on the round duct where it drops down from between the joist, then running the rectangular duct up against the basement's ceiling joists, which should be relatively easy and inconspicuous to run across the ceiling and then retie it into the round duct where it goes back into the ceiling joists. You can then frame a box around the duct and finish with drywall. There will be a little air flow disruption, but shouldn't be too bad.