An important stability consideration when probing currents is whether moving the sensed cable alters the reading. For instance, if you approach a current carrying wire with a Hall probe, the gain depends on the spacing.
A Rogowski coil solves this problem by using a symmetrical geometry (torus) in a uniform magnetic medium (air), and outputs a voltage.
A current transformer solves the problem by using a ring of very high permeability material, and outputs a current. Any geometry error is 'diluted' by a factor of the permeability.
You can have one or the other, they are mutually exclusive.
You can bring some of the Rogowski thinking to improve a current transformer by making the core and sense coils symmetrical, this will reduce geometry errors.
You cannot really bring any current transformer thinking to improve a Rogowski coil, you just end up with a half-designed current transformer.