Hi -
There’s several threads about CPI and Bluetooth [link here]
I am not an specialist in BT modes, but I have used a MOCUTE BT game controller with CPI and it works. BT is a thorny issue and compatibility between varieties is not guaranteed - not even in high end mobile and audio devices.
Whilst I fully understand your point that GPU may have been advertised, I did not buy my CPI with this in mind. I bought it because it can be hacked and modified and runs classic console games in a small portable format.
{disclaimer: I am just a customer and do not have any stake in CPI.}