Update: one day after an unsuccessful attempt at solving my issues with a screen sharing session, my problem “spontaneously” vanished. I suspect that Elite Dangerous received a software update that made the game load freetrackclient.dll properly (since it did not do so during the support session). I now have glorious eye tracking in ED:H!
A small thin tarnishes the experience and that is that when centering the viewpoint using a hotkey set in ISE, the view is not perfectly centered. But I believe this is a know defect that will be fixed in th next version of ISE.
Mad props to the Tobii support staff (Emelie, in my case) and Henrik for all the help!