Cheapest way to simulate a Bluetooth mouse/keyboard

I know that you can use a certain Arudinos such as the Arduino Micro to simulate a USB keyboard. I am looking for the cheapest way to do this over Bluetooth instead of USB.

The typical Bluetooth modules such as HC-05, HM-10 etc only seem to support serial communication and can't simulate a device such as a keyboard or mouse from what I understand.

Are there any other easy-to-use cheap chips that can simulate a Bluetooth keybaord and allow me to make it send certain key actions using an Arduino or the like?

The cheapest way I coudl find so far would be to buy a cheap actual Bluetooth keyboard, rip out the PCB and solder an Arduino to it...

(The solution needs to work on any system that supports Bluetooth keyboards. Android, Mac, Windows etc... I'm mentioning just in case someone wants to suggest me to use an Arduino Micro and two HC-05 modules to have a kind of fake Bluetooth device.)

If your desperate to keep it cheap (but not simple) then you can flash the HC-05 module with the RN-42 HID stack.

Thanks, I will give that a try. Also, I found out that I can use an ESP32 for this (they are about $3). In that case I wouldn't even need an Arduino.