How do I set up an Arduino to be a wireless HID?
I'm not sure for Arduino, but it has been done with ESP32. If you have not yet spent the money into a Bluetooth module for Arduino, you'll find that the ESP32 is not expensive either.
You can find the code on Github. The author uses a TinyPico because it's pin compatible with one of his earlier projects. IMHO, any ESP32 should do as well.
The key thing is to
#include &lt;BleKeyboard.h&gt;

the ESP32 BLE Keyboard library that you need to download and install. That library comes with example code that should be easy to adapt to your needs.
You can program the ESP32 in Arduino IDE, but you need to add support for the boards by adding a URL to the board manager.
Have fun with the project. It sounds very interesting. Maybe I try that myself.