If I understand correctly it would flatten the hysteresis curve. So it would allow to store more energy and it would also decrease the permeability of the core. Would it have another effect or will some type of compensation be needed in the system?
Just thinking this through, without having done it: it would cut the inductance per turn roughly in half. So to store the same energy you'd have to increase the current by \$\sqrt 2\$, or you'd have to increase the number of turns by \$\sqrt 2\$. Actually bringing the core up to the same value of H as before (and thus using that "extra" energy storage in the gap) would require roughly doubling the current, or you'd have to double the number of turns.
Assuming that the coil winding volume is already full, just doubling the number of turns would increase the winding resistance by a factor of four. Similarly, increasing by \$\sqrt 2\$ would double the winding resistance.
So you'd have a definite trade-off between the energy lost to switching and/or coil resistance vs. the energy the coil could hold in each cycle. I suspect that for any given size of core, and assuming copper wire, there's an optimum that depends on the switching frequency, and the capabilities of the switches you have available.