You'll probably be able to compensate for small metallic grounded things in the pitch field. The main issue is getting metal near the coils and having that mess with the inductance / damp oscillation. I've seen a metal alligator clip connected to close to a ferrite choke kill analog oscillation (it seems the long leads are there for a reason).
The 20x4 LCD general operation doesn't seem to disturb anything, but I saw some minor interference when I was updating it at the encoder sample rate of 12Hz, so now I only update it (again, at 12Hz) only when the displayed data needs a refresh. Your idea of doing this at some integer sub multiple of the sampling / synth rate is a good one, and is probably the best you can do.
If you have an audio DAC you can gain access to in your Theremin, try running the pitch axis number out of it and recording / monitoring / analyzing that with audio software. That helped me a lot early on to squash internal interferers. I found that pre-processing the number with a 4th order 10Hz or so digital high pass filter was necessary to get rid of DC, which allowed easier analysis (this filter doesn't have to be very scientific or fancy, it could even just be a cascade of first order filters). You can learn a lot just listening to the resulting signal, and an FFT will reveal even more.
One other surprising interferer was from the audio synth, the FPGA logic draws quite variable current depending on what's going on with the audio processing, and this made its way back to the oscillators (in my system the FPGA is an integral part of the oscillators). I killed that by making the axis number low pass filters variable, the cutoff of which dependent on the output filtered value.
