Using an op amp to balance a Wheatstone bridge by heating a sensor wire
The issue I'm running into is that the op amp is unable to provide
  enough power to heat the sensor wire


Based on your circuit (and values) you're going to have a more basic issue. That circuit has only one stable point and that is when the op-amp is delivering 0 volts at the output. In other words you need to build in what is known as an "over-temperature" so that Rs “runs” warmer than ambient temperature by a known margin. 

This entails adjustment of typically R2. And, you need to know the "over temperature" because, to calculate the thermal conductance (or resistance) of the water (or air or other medium), you need to know how hot Rs is running compared to ambient temperature.

Then, you'll have three stable points of operation - the first one is when the op-amp is delivering 0 volts (unwanted), the 2nd when delivering a positive voltage and the third when delivering a negative voltage.

So, what you need to do now is force the op-amp so it can't deliver stability at 0 volts or negative voltages and, to do this you simply run the op-amp from a single positive supply because the output can never get to 0 volts or go negative.

Then you have something like a working system and adding an emitter follower current booster is almost trivial in comparison. I've done it using emitter followers and inverting common emitter amplifiers but, with CE you have to invert the polarities of the op-amp inputs. The circuit works well and delivers good performance if you are looking for accuracy: -



The inverting input above is not wired to the emitter on the circuit you need. You still wire it to the junction of R1 and Rs. All the bridge resistors connect in the above circuit replacing \$R_{LOAD}\$. The non-inverting input connects to the junction of R1 and R2 as per your original circuit.