Thanks for the suggestions!
I've revised the schematic like so, and added a jumper for a small NTC temperature sensor if needed.
In addition, I added a low pass filter on the output of the LM399 (Is this worthwhile?), and also removed the trimmer stuff. (One lm399 is like ~7.001V which makes it close enough to 7V that I don't need to do much trimming)
The original idea was to ovenize just the resistors (and possibly the op-amp) with a PTC heater, then encapsulate everything in some insulative foam.
The LM399 will be placed separately.
Unfortunately I do not have another 0.1% ~3k resistor to spare, so I'll use 3 0.1% 1k instead. (Originally I wanted to parallelize 3 10k, but I don't have that)
For the buffer, I think it doesn't need to be high current? I only intend for it to be a voltage reference for testing purposes.
As for the rectifier, I've decided to use a discrete module, which seems to take AC 18V and turns it into DC 24V. I may possibly need to find a small enough transformer to put into a case.
A few questions:
How can I know the current flowing through the zener diode based on the resistance?
Also, why should U4 be located near the PTC?