Hi, I am having connection problems with the new Shelly BLU Button RC 4.
It was discovered and added without any problem, but Home Assistant recognises the button presses very unreliably, perhaps only every fifteenth or twentieth button press is recognised.
My setup ist Home Assistant Core in a VM on Proxmox and Amazon Bluetooth 5.3 Adapter.
This works perfectly with several BLU Button1 buttons and other Shelly BLU devices.
The Shelly Smart Control and BLE Debug app register every button press of the BLU RC Button 4, so it seems the problem is in the Home Assistant or BTHome.
I don’t use a Shelly as a BLE observer in HA because my flat is quite small and the adapter’s antenna reaches everywhere. And I thought it would be easier to set up.
It also make no difference if the BLU button is next to the atennea or few meters away.
Unfortunately many of the USB bluetooth sticks are built to work on very specific OS/Driver/HW combination. I went trough so many bluetooth sticks before landing to a golden-standard which just works with Linux and with every port in my machines.
You will most likely want a device with RTL8761 chipset.
Go with the EDUP with external antenna (or any other with RTL8761 chipset and external antenna)
(oh, I just noticed that the adapter you have has the same chipset, there were some discussion about the firmware being different as your device is stated to support bluetooth 5.3, not 5.1 where the chipset was originally made?)
And bare in mind that many usb3.0 ports can cause interference with USB sticks, so you can experiment with usb extension cables to both get rid of noise near the port at host machine and to better placement of the antenna.
Make sure your 4 Button device is “paired”. I had the same issue/question as you but after futzing with it (use Shelly App to update firmware/pay attention to button pressing sequence requested from the App), then the device came up in BTHome integration and every button was addressable/programmable.