Could it also be used to bridge a BT keyboard to USB for the semi-rare cases where having a wired keyboard is necessary (BIOS access, boot loader, login screen etc.)? I expected this type of product to exist, but couldn't really find a good one after a cursory look.
I don’t know of an off-the-shelf way to implement what you want. There’s a usb-usb converter in QMK which allows you to add QMK features to any keyboard, but QMK does not have Bluetooth support. ZMK is designed for Bluetooth but doesn’t seem to have anything like QMK’s protocol converters (tho I might have looked in the wrong place).
You might be able to DIY it with a Raspberry Pi Pico W, by gluing together the picow_bt_example_hid_host_demo and tinyusb_dev_hid_multiple_interface examples from https://github.com/raspberrypi/pico-examples
In principle, yes, but you'd need something additional to act as a Bluetooth host. You'd probably be better off using a part that can do both, like an ESP32.