Is it an option ? I know your controller connects but its not supposed to connect through your systems bluetooth thats not how sony bluetooth works. Wow analogs are triggering physical buttons bro ? you definetly did not pair with pc then. i cant remeber its been a minute since i had ps3 but i dont think it had the reset button It would not pair my ds4 controller with my ps4 address on it, but like i said dualshock has reset button that clears it.
The app can actually pair my ps4 controller now but it required it be reset and i dont think a ps3 controller has that button on it. it has to be an identical mac address as device your syncing too for it to sync like it does on your sony. i dont know why your right dpad is slow, did you change the ps3 mac address with sixaxis pair tool on pc ? that step would be crucial. Additionally you gain access to features like analog polling and remapping/disabling from within the app. You cant see it on a ps3 controller because it lacks an indicator like the dualshock 4 lightbar but its the same tech. The controller takes your system bt mac address and initiates true sync.