Thanks everyone for your input and, well, sorry for the delay. Obviously, every new tip is welcome and I am ready to start thinking through the backhand, starting almost from scratch if necessary.
On the other hand, I just want to precise a detail: I will not change my grip for the heck of it... I use a pretty standard eastern backhand and since the habits are hard to fight and that I could get to hit well with it, I would rather focus on what my body does than on how I grip the racket.
Of course, drills or links to videos are also welcome. Continue to add to this and maybe even make it a backhand discussion for all of us, trying to master the forgotten art.