Maybe I'm having a brainfart, but I can't see why the ref pin can't go to -Vs in this case, as it shouldn't exced the common mode range of any of the op-amps inside the AD8236.
I was having a brainfart.
Look at the internal schematic of the op-amp. Suppose the input is 10mV, the output must be 873mV. The golden rule to remember is the op-amp will keep the voltages at both inputs equal. If pins 1 and 6 are at 0V and pin 4 is at 10mV, the voltage on both of op-amp A's inputs must be at 0V and the voltage on both of op-amp B's inputs 10mV. The voltage across R
G must be 10mV. In order to get 10mV across R
G, op-amp A's output needs to go negative. No current can flow through op-amp A's 210k resistor, as it's got 0V across it, so it has to flow through the 52k5, which is connected to op-amp A's output, which must be negative.