I got a Beloader working directly with the Quadstick and the PS5. No other hardware.

[QuadStick]

A post in the Facebook group prompted me to try it again and I got it working after updating the Beloader firmware.

The quadstick needs to be in emulation mode 3 or 7.  Boot in PS4 USB mode doesn't work with it.   3 is XBox 360 and 7 is Dualshock 4.  7 is what the Quadstick switches to when it detects a PS4 console. 

Here's the Beload tutorial video:  https://youtu.be/88rNyBVnsCs

You can find their website in the video description.  Setting up the Beloader is not simple.

You will need to modify your config files with a Preferences sheet to run in mode 7 if you want touchpad.  Also, turn on "Titan two PS4 flag", or set the preference "titan_two" to 1 in a game file, to prevent the Quadstick from resetting the connection every 7 minutes.

DO NOT MAKE THAT YOUR DEFAULT.CSV OR PUT IT IN PREFS.CSV.  Emulation mode 7 has no flash drive access for making changes to the Quadstick.