Will the cold water from a faucet be warm if a re-circulation bridge valve is used?
Too much over thinking.  The set up the OP is describing is that the pump turns on when the hot water is turned on in the master the master.  Once the water starts to warm up, the temperature bridge closes and pump shuts off.  The bridge closes before the water gets too warm.  I have a re-circulating hot water pump, and the water out of the cold faucet never feel hot or warm.  It is just not &quot;cold&quot;.  So if you are looking for &quot;cold&quot; water out of the faucet, then a recirculating would be out of the question.
No- the water will NOT be hot coming out of the cold-side faucet, just not cold or cool.
If you think about it,  The water out of the cold side of the faucet will be the normal temp of the cold faucet water until the existing water on the hot water side has been expelled and heated water has reached the bridge.
If you want to spend the additional money for a separate return line to the pump to have cold water to drink, the builder will most likely accommodate you.