Hi Charles-Henri,
Is there a way to configure it via the usual customizing options?
No there is not. The Table T706Z1 is the basis for the customizing options and the field control possibilities.
If you want to adapt more fields for FITVFELD / FITVFELD_WEB then using SE16 is the feasible way to do it, in several SAP Notes this is also the approach to adapt certain new fields to the field control.
One word of warning from my side: The business logic in the frame programs does not handle the entire T706Z1 generically up to 100%. I.e. it is unlikely but still possible, that if you add certain fields in T706Z1 and configure them in FITVFELD / FITVFELDWEB, the configurations might not reflect. This part is trial & error.
Cheers, Lukas