Dardick failed in the market for a reason.
Applied to an aircraft cannon, the rotating chambers of the Dardick would potentially have some use.
Also, it utterly failed to make any constructive use of the 65% increase in effective magazine volume that trounds fill and circular rounds don't!
...
Anyway, as for the concept, it sounds good so far, but I have to pick on the gas system. I think something similar to the AK's gas system in operation, but not appearance, would be more conducive to your goal of light weight.
Granted, AKs proper have huge bolt carriers that resemble something that's a felony to sell in certain parts of the South, but imagine a bolt carrier that works like an AK's but looks a little like an AR's. Since it's a bullpup, and action length behind the magazine is not desirable as it leads to excessive length of pull, the bolt carrier is a short, minimalistic affair with the rude, radial lugs of the bolt sticking out the front like some sort of demonic dremel attachment, the camming surfaces carefully machined into the carrier body, a piston rigidly attached up top, and not a whole lot else.
The piston resembles the gas key on an AR-15 bolt, except that instead of being a female attachment, it's a solid piston.
Gas is tapped either from a port in the barrel, or from a gas trap at the end of the muzzle and sent breech wards down an AR-style tube of minimal weight and bulk. This tube expands into a strong, rigid piston housing that the piston rides in. The gas forces the piston back. The piston is welded to the bolt carrier. There's a recoil spring somewhere. This is the same system that's been running AK's for 61 years now, and it works just as well now as it did then.
Err... there are some holes in the AK gas system that I can't really explain, but I think my explanation is more or less correct, and that even if it's not, such a system as I describe would work.
The advantage of putting distance between the piston face and the site of the gas tapping, especially if you use a gas-trap is that it will slow down the cyclic rate of fire, and allow the propellant gas to cool adiabatically, which should keep fouling and parts wear down.