It is customary to use a globe valve to give the regulating capability required for a bypass, and the Cv values you have given are typical for globe valves.  A gate valve is more usually used as an isolation valve as it does not have a good characteristic for regulating.  The 1/2" valve may have a slightly lower calculated Cv than would normally be expected for that size - but it just means the valve will not be used fully open.
Your scaling rule seems to come from the same basis as Equation 3-16 from Crane TP410, but the Crane formula is more complete.  In Crane the "K" used is the same "K" as is used in flow resistance calculations for pipe fittings (we have had that discussion!!) and it is
Cv = ( 29.9*d2 ) / ( K0.5 )

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