Just an update about my “situation”.
I have finally bought a NES30 Pro pad.
Linked in bluetooth, I have played around 30 minutes to Street Fighter 2 turbo on snes : no issue.
I have also played 15 minutes to Thunderforce 4 on Megadrive which usually is an issue : no problem.
Of course I need more time to confirm. But I am more and more thinking this is an issue with the PS3 driver and not with the bluetooth.
As e a reminder, I met same issues on both PI2 & PI3.