Signal conditioner for a piezoresistive pressure sensor
The strain gauges in this circuit are not used in a constant current arrangement, but are supplied by a constant voltage source (Vb) instead. I think the 1mV/psi response you mention has been determined for a supply voltage of 12V (which you also mention in your question). 

Since the bridge is operated in a constant voltage configuration, if you supply the bridge with something other than 12V, it's voltage response will be proportional to the supply voltage by the factor Vb/12. 

Said differently: the current in each leg of the bridge will be nominally Ih = Vb/(2*R), where R is the nominal resistance of each bridge resistor. If the strain changes only one bridge resistor by ∆R, then the voltage response of the bridge would be v1-v2 = Vb/(2*R)*(R+∆R) - Vb/(2*R)*(R) = Vb/(2*R)*∆R, so it's obvious from this equation the bridge voltage response is proportional to Vb.

Incidentally, the bridge offset (1mV @ 12V) will also be scaled by Vb/12.