This is not related to a Bluetooth problem but key mappings
I'm not sure Bluetooth is entirely out of the equation yet, but once again I'm not very familiar with this protocol.
The controller can pair, but once at the binding screen, all buttons but two work.
My understanding of the binding tool is that it waits for an input and binds it to the key being displayed. As pressing "Back" and "Home" doesn't cause any reaction, I'm wondering if the binding tool actually gets an input.
The other option is that the binding tools waits for a specific range of possible inputs instead of listening to everything, and maybe these 2 buttons are somehow out of range.