DGT boards are surprisingly expensive even when broken.
Here is a high-level overview of how they work https://www.chessprogramming.org/DGT_Board
I think the piece simply contains an inductor and a capacitor. The values are different for each piece. Soldering SMD inductors and capacitors back to back would make the piece part tiny. Something like they do for wireless LEDs (https://www.aliexpress.us/item/3256804431028018.html).
The amount of coupling to the receive coil would depend on how close to the transmit frequency to the resonance of the LC circuit. So, the scan involves trying all the frequencies in each transmit coil and sampling the response in each receive coil. The algorithm described there says "measureFrequency", but I don't think they actually measure the frequency. It is more likely that the set of frequencies is fixed and they measure the amplitude of the received signal.
They also mentions scanning 35 pieces, but I'm not sure why you would need to differentiate individual pawns.
It would take quite a bit of experimentation to tune all of that, so it may be easier to try that with a signal generator and scope at first. After that works, then it would be time to think about standalone circuit.
I already found this page but it doesnt seem to have much more information than what is described in the patent. Although there is a picture of the transmit and receive coils printed on a polyester film. But are you sure that there isnt another way of measuring the resonance frequency except for looking at the amplitude of the received signal of a set of frequencies applied to the transmit coil? The text of the patent states the following:
"The operation of a device according to the invention is as
follows. Control device 11 selects a transmit coil 18 and a
receive coil 19 (see FIG.3). Transmit coil 18 is subsequently
connected to the output of an amplifier arranged in control
device 11 and receive coil 19 is connected to the input
thereof. A resonance will occur in the circuit due to the
induction between transmit coil 18 and receive coil 19.
Because a playing piece 16 with a resonance coil 15 is
Situated above the interSecting part of transmit coil 18 and
receive coil 19, resonance coil 15 of piece 16 will influence
the above Stated resonance Such that a fixed resonance
frequency results."
It doesnt sound to me that the solution involves needing to apply different frequencies but that the circuit just oscillates at the frequency of the resonance coil but i might be wrong and the patent just isnt very clear about the exact process.
This is all just for personal use but i think it shouldnt matter since the patent is expired anyway.