Okay, it looks to me like you're just using the DAC as a voltage reference as far as the front end goes. You're exciting the half-bridge with the (noisy) power supply. Yes, this is sub-optimal.
It would be better to have something like a Wheatstone bridge, but also to excite the bridge with a reference voltage rather than the power supply. Consider the below circuit. The 3.2:1 amplifier is adapted from the recommended reference circuit for the ADC. If you change the amplifier part number, it may not be happy with C2 so other changes may be necessary.
simulate this circuit – Schematic created using CircuitLab
You might want to buffer the outputs marked "To ADC" with a dual zero-drift op-amp**. If you are concerned about noise, the op-amps could be configured as a fully differential MFB filter, but that's probably not necessary since the signal levels are pretty high with a thermistor. You do have large currents floating around with the output driver, so layout is very important.
In particular keep the grounds for the R2 and Rth together with each other and well away from any high current circuitry.
You could use a network for R1 and R2, there are some nice parts available that track very well. You are looking for ~5ppm stability which is likely with two 0.1% resistors off the same reel, but you can have it guaranteed for more money. Of course the reference resistor is the most critical since it is being compared with the thermistor so the absolute resistance stability matters rather than just the ratio, so put your money there first. Even better, put it in the controlled area so temperature variations don't matter so much (but stability still does).
** The delta-sigma converter you are using does not have built-in buffer amplifiers and the input impedances (common mode and differential) are relatively low and vary with clock rate. See SBAA086 for more information.