Mark-

The light weight of this gun is a fair bit of what makes it such a sweet-looking project (to me)

A light-charge (short chamber, as Peter said) 20-bore would be sweet.
Plenty of power (even at 2-1/2 drams) to kill deer and elk, it'd make a sweet and handy stalking rifle, and the outside diameter of the barrel wouldn't 'look out of place' in relation to the bore diameter.

Snyder - my concern would be the resulting weight of the rifle once you got that 577 barrel tucked in there.
The barrel-blank *could* be turned thin-wall between the chamber and the muzzle to save weight, you'd want at least a partial section of full-bore (to the shotgun tube) somewhere in the middle to keep stress low at the muzzle...

If that existing tube was really done, with no hope for service - I'd go 20-bore if I could.

Cheers
Tinker