This is honestly reminding me of the upgraded revolver from the original bioshock.A similar idea to the chain gun used in a more modern revolver was the Dardick revolver. I have had similar ideas on a steampunk revolving rifle. But I would try to increase ammo capacity through the use of a cartridge magazine combined with a mechanical auto ejector. Which others have doen in OTL. I will share the examples I know of in order of my own personal discovery of them.
There was a guy who shared his progress online a few years ago of him converting a Ruger revolver with a Spencer-like tube magazine and gas ejection. He called it the FrankenRuger.
A similar idea was done over a hundred years ago by a guy named Chaine with a modified Webley revolver, but with a mechanical ejector in the form a a finger attached to the trigger. As well as what seems to be a magazine/buttstock in the form of four tubes that spin on an axis
A much simpler mechanical ejector was patented by Silver and Fletcher in the 1880s that Forgotten Weapons made a great video on
Surprisingly the most efficient form of mechanical auto-ejecting revolver that I know if is also the earliest. With the Swiss made Von Steiger revolvers that I learned about through Forgotten Weapons. Unlike in the Silver Fletcher video he shows how well it actually works.
So I think that the Silver Fletcher design is perhaps the most to mass produce, but I'm not sure how well it may work. I also think the Silver Fletcher design could even be incorporated into Richards-Mason style conversions rings used to convert percussion Colts and Remingtons into cartridge firing revolvers.
Another good source of inspiration for Steampunk guns that I know of I are the animations made by GarethWasHere