Why subcontracting a redesigned Barracuda would not be cheap

Would having Boulton Paul build a modified Barracuda with a different engine be a cheap way to improve the aircraft?

No; proprietary tooling, engineering differences, and limited government authority would make the transfer costly and politically difficult.

Moving a Fairey design to Boulton Paul involves more than transferring drawings because each firm used its own equipment, production knowledge, and engineering practice. A redesigned aircraft might improve performance, but it would not automatically be quicker or cheaper. A new team might also fail to identify the design faults sooner than Fairey's own engineers. The Ministry of Aircraft Production had enough power to impose such arrangements mainly from late 1940 to 1943, and even then political consequences constrained it.