I think that Nissan should sell the GT-R as it is, and Infiniti should sell a sedan with a GT-R-based chassis, GT-R engine, four doors, rear-wheel drive, a choice between a six-speed manual and a dual-clutch, Infiniti ride quality and sound deadening, and a more traditional (and more luxurious) Infiniti interior and exterior.
The GT-R would be the fast car for 30-something computer programmers who grew up playing Gran Turismo. The Infiniti wouldn't be capable of the blazing-fast Nurburgring times, but would steal customers away from the M3, RS4, and C63 AMG.
And like you said, that would spread the cost around and make both models more profitable.