"I just received a new FPGA board from Waveshare, and now both encoder modules are working as expected ;-)" - sarob
Yay! Of the 45 or so boards I've received from Waveshare, only one was bad, and to their credit they replaced it even though it was somewhat out of warranty. I appealed to them that the warranty period wasn't clear - which is true! - so I didn't think I needed to power it up immediately upon receipt. Just obviously a brick, even though the regulators, oscillator, etc. were functioning fine.
"...in the short run I was wondering if it might be possible to test basic functionality using an off-the-shelf inductor."
Theremin's curse: there are few off-the-shelf components that are ideally suited for use in a Theremin. It's only gotten worse with the death of tube radio, because old pi-wound RF chokes aren't necessary anymore, and the vast majority of the chokes which remain have been miniaturized to the point of uselessness for our purposes. It's very much like trying to find a commercial choke to base a Tesla coil around - you similarly need high Q and low self C (high SRF), and a physically large-ish (by today's standards) build that separates the terminals. I'm not saying it's impossible to find something that might sorta work in a pinch, but it will likely be low Q and drift a lot with any current through it at all if the wire is too fine.
Here's an old version of the 4600 series Bourns / Miller choke datasheet: https://d-lev.com/research/4600_series.pdf. In somewhat later versions they gray out most of the lower part of the table and tell you not to use them in newer products. The Bourns spec of SRF @ 2.6MHz seems OK (you need the SRF to be comfortably above the operating frequency, for the D-Lev the pitch side runs around 1.2MHz, the volume side around 800kHz). The Q is specified as 83 @ 1kHz, hard to know what it might be at 1.2MHz. But you would need 10 of them in series for the pitch side, and 20 for the volume side, and at $6.54 a pop you're talking rather serious money for something you could wind yourself for a few bucks, and which would be much more ideal for this application.
These folks claim to have a $12.50 2.5mH choke: https://www.ctrengineeringinc.com/rf-chokes-and-inductors/
Searching for "pi-wound choke" turns up some older chokes.
If you have an inductance meter (I recommend the inexpensive resonance types) you could try scramble winding some fine wire around a ferrite core, perhaps in separate donuts, until you hit the mH target. Then play with the AFE C divider until you get a healthy enough signal to lock. But if you're going to that trouble why not just bite the bullet and make the real thing?
[EDIT] And there's a possibility of me winding you some coils. Do you live in the US?