Use bluetooth keyboard to access/edit BIOS?

A keyboard using Bluetooth cannot access the BIOS. Logitech Bluetooth keyboards get around this by having a dongle that pairs with the keyboard in a more basic, non-Bluetooth mode until the driver kicks in and switches modes. Microsoft might be similar mode with their keyboards and dongles, but I cannot confirm that.

Try using a Bluetooth adapter that supports "HID Proxy" mode.

According to Microsoft, "some" of their keyboards work in dual mode:

Devices with Bluetooth wireless technology generally do not work until the operating system is loaded. However, some Microsoft products start up with 2.4 GHz, and then switch to Bluetooth, so you can use your keyboard and mouse before the operating system is loaded.

If you're happy using a dongle, you might as well just use the proprietary 2.4 GHz protocol. But if you have a laptop with built-in bluetooth you can save a USB slot. Use the dongle when you need to access the BIOS, but use the built-in bluetooth normally. This would work great, as long as you don't lose the dongle!