I had a pretty bad go of it with this controller. I moved to the ps3controller driver and I did not get that to work per the docs. I tried to sort out which clone I may have, using lsusb, evtest, and dmesg but none gave me any indication other than Sony. So, I tried them all-- Official Sony, shanwan, and gasia. Although I appreciate the info to get additional bluetooth debugging, that project is "crossing the line" with regard to the time I am willing to invest.
Perhaps to make you laugh a bit-- I used retropie for several years with a Logitech F710 controller and an Intellivision Flashback controller. The F710 was getting worn out so I bought another one. It worked terrible! While the new F710 connected no problem, it was very clunky in terms of usability--buttons didn't work quite right, sometimes the control is sticky. Odd! Years ago I had a PS3 and I remember the controller fondly so I thought, I'll try one of those. Needless to say, instead of playing games I'm troubleshooting why this apparently poorly made clone isn't connecting. While the "thrill of the hunt" for the solution to the problem was fun, I'm done.
So, I'm not quite sure which controller I will try next, but the PS3 dualshock clone just found its way to my electronics-recycling box while I sort out my next step.