Can disconnecting the cold switch on a shower pump save money?
To answer the question in the title "Can disconnecting the cold switch on a shower pump save money?"  The pump you linked to uses 345 watts.  Assuming you use the cold water only in the bathroom for 10 minutes a day - cold tap, toilet flush, then you're looking at 5 hours per month at 345 watts.  Call it 2 kilowatt hours.  My local utility company charges about 15 cents per kilowatt hour.  Your net savings will be $3/year.