How to make LT1168 instrumentation amplifier work with single supply?
Your main problem: -


  Wide Supply Range: ±2.3V to ±18V


This means that the minimum single supply voltage is 4.6 volts i.e. twice 2.3 volts. Trying to run it from a 3.7 volt battery is always going to cause anguish. Even if you suppied it 4.6 volts you have to be aware of what the output can deliver under load. Take a look at this in the data sheet: -



If you tried to drive a load that took peaks of (say) 10 mA, the output voltage range is 1.3 volts to 3.3 volts.

I think you have chosen the LT1168 without too much thought.

Of course you could always boost the 3.6 volts up to (maybe) 6 volts or more using a boost converter.

Pin 5 (reference input) would normally be provided with a mid-rail voltage. This can be achieved with a resistor potential divider. Use low value resistors (i.e. 1 kohm) because the ref input has an impedance of 60 kohm.