How do you know the valve is shot?
From your use of the singular, I gather you have a 3-way valve (one) in the cold water line and nothing but a "T" fitting in the hot line.. Two possible issues.
1: You are getting only warm water and thus assuming the bypass valve is letting cold water bypass.... Suggestion: Check the outside shower to make sure you have not left the mixing valves open and cut off flow at the head (Iniside shower too for that matter)
2: no how water flow: This is USUALLY the fault of the outlet check valve, not the bypass vavle.. It is harder to replace.. (Some folks have been unable to remove it from some heaters)
One "young lady" I know simply pulled the guts out of it using needle nose pliers.. Worked great since she winterizes by the dry method (Air not pink stuff).
She fixed it later.... (Bought a new trailer)
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times