After a bit of searching and help, I found out how to use my 8BitDo Ultimate Pro Bluetooth/2.4Ghz controller in 2.4GHz mode in the Switch mode, including with full gyro support. Note: there are two firmware versions, one for the controller and one for the receiver. This is only possible with a specific version of the receiver firmware, V1.02_Beta1, and I've only tested it with a specific controller firmware, V1.03.
Download
The Reddit user koloved has uploaded the files to their Google drive. Download both to make sure your controller is running the right version as well. This includes both the installer EXE and the firmware you need. Unzip them both.
By the way, if someone knows a good place for me to store those files long-term, let me know and I'll upload some mirrors.
Install
Controller
Although this step might not be necessary, it's a good idea to make sure the controller itself is running the right firmware. You'll need to plug in your controller directly to your computer. Then open the UltimateBL103 folder, run the 8Bitdo_Update.exe file, and select the firmware in the folder labeled Ultimate_Firmware_V1.03.dat when prompted. Easy peasy!
Receiver
This is the crucial step! The USB 2.4 Ghz receiver has its own firmware that we need to update. Run the UltimateBLRR102B1m folder's 8BitDo_UM_BT_Receiver.exe file and follow the steps. When prompted, select the included UM_BT_Receiver_V1.02_Beta1.dat file as the firmware version you want to install - it might warn you that this is an outdated version, but you can just hit OK to proceed.
Activate
Now that the firmware is installed, it has to be activated properly. The controller can operate in a few modes, but there's only one mode that seems to send gyro over 2.4 Ghz. Here's how to activate it:
Turn the controller off completely and disconnect the receiver
After a few seconds, connect everything and turn the controller on. Ensure it's in 2.4 Ghz mode on the back
Hold select (the minus button left of the star) + A for a few seconds. This puts the controller in a special Switch mode. Tip: sometimes the normal Switch mode (select + Y) works as well, so try that if necessary
You should hear a disconnect and reconnect sound from Windows. For me, the controller then disconnects itself again after a few seconds and has to be switched back on - not sure why, but it doesn't seem to be a problem
The controller should now be identified as a "Nintendo Switch Pro Controller" and everything should work!
If it doesn't, unplug and replug a few times, reboot the program you're trying to use, etc. It can be a bit finicky, but once it works it seems to work forever
Dolphin Emulator
I'm using Dolphin Emulator to play Wii games, so here's a quick guide on that. To use the controller, you want to use the "Emulate the Wii's Bluetooth Adapter" mode with "Emulated Wii Remote" selected. Then in the Configure screen, make sure you select the "Nintendo Switch Pro Controller" input.
To verify motion is working, go to the Motion Input tab and move your controller. The red and blue dots should swing around the circle as your controller moves. If that's working, you're good to go! Otherwise, try the steps above again and give it a few restarts.
So glad it helped! The gyro might work over Bluetooth when the controller is in its normal Switch mode. The great thing about using it in 2.4GHz is that you can easily toggle between a Bluetooth device, like a Switch, and your computer. I hope it works for you!