I'm building a PI controller with inputs read using instrumentation amplifiers (for sensing resistors and floating grounds), rather than subtracting the two input signals with a third chip, is it possible to use the reference pin (as shown in the circuit) in series to accomplish this?
Your circuit is somehow weird. The purpose of the instrumentation amplifier is to have very high input impedance at the input stage and a good common mode rejection ratio. In your case you have some non symmetric components at the input stage, that will result in bad CMRR, extra you have a resistor placed in between both inputs, why?
The Vref needs to be connected to a low impedance, so connecting an opamp output to Vref input is a way to go. But there is one another strange thing: you have one measurement as Setpoint and other named Sense, in control theory you would need (Setpoint-Sense), what you'll get is (Setpoint+Sense), OK you can still swap the Sense inputs to get negative value.