I've already received lots of feedback elsewhere on how to improve this project.

The first and "easiest" thing to change was the grounding arrangement for the heater. Currently, the heater and reference share a ground. This is problematic because the current from the heater could lift the ground of the reference. To fix this, I cut the ground trace on the emitter of the heater transistor and ran the ground separately back to the power supply.

The second thing I am going to change is to have better control of the temperature. Although I did not see any oscillations in temperature initially, after making the grounding changes above I now see oscillations. For a revision 2 I plan to add PI control of the temperature. I also plan to use two resistors: one mounted on top as it is currently, and one mounted below the board. This is to even out the temperature gradients and remove thermocouple effects. I also plan to use a smaller temperature sensor with less thermal mass and resistance, and mount it directly to the reference instead of to the resistor.

Fun note from the MAX6226 datasheet:

"Pin 8 must be clear of any mechanical and electrical
contact. Neither copper nor solder/paste mask must
be placed underneath its land pattern"

I did not know about this..