Strange output from an AD595 in single supply mode
This turned out to be caused by a few problems, most of which I could rule out by proceeding down the stack.


The Arduino's input pins are naturally measured against the GND pin on the board; I was overzealous in isolating the amplifier supply and ground rails. This explains why all the digital readings had a low ceiling.
The filter schematic flips the IC around the X-axis compared to the data sheet and other schematics in the same guide!! The decision to do that boggles the mind. My filter was thus attached to the wrong side.
The "negative" (according to the thermocouple and IC manufacturers) pin is on the upper-left side (pin 1) of the IC. However, the common ground must be attached to the "positive" temperature input, along with the "positive" thermocouple wire. This plus issue 2 led to a lot more wire swapping than I would like to admit.


Below is a working (though still uncalibrated) example of two AD595AQs, one with the RC filter in front, and one without (connected to probes). (Ignore the melty wires; they were tested for resistance and are definitely not on fire right now.)