I think a BT transmitter board would fit the bill for both you and the OP, and could be mounted and hardwired into a game in a way that appears "stock".

I used an identical module as the one in the link in a shortwave receiver with good results (wanted to use noise-cancelling HPs), and I don't see why it wouldn't also work well in this situation. It requires +5v supply and can take a line-level analog signal from the Pinsound adapter, and features pins for a momentary SPST pushbutton switch and LED for BT pairing mode.

https://www.amazon.com/Transmitter-Bluetooth-Wireless-Speaker-Headphones/dp/B081TD27G7/ref=sr_1_20

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The tiny size is a feature, not a bug! Very easy to unobtrusively mount just about anywhere, though I agree the nonstandard header definitely earns a zonk (I never found an adapter so I ended up simply hardwiring everything in my radio project).

The only things you need to solder on this board are the connections, all of which are through-hole and nicely pretinned, so it’s definitely NOT beyond your skills if you can work on B/W boards!

Ha, I guess it was late and I was half-conscious, lol. Yes, I remember being amused by the gnashing of teeth (not only in pinball, but also audio and other DIY electronics) and the hyperbolic posts suggesting it would "kill" the hobby.

Thanks for the link, now I want to experiment further with these boards.