A isotropic, simplified subset of this menagerie.
Orthogonal movement: none, wazir, rook
Diagonal movement: none, ferz, bishop
Knight jump: no, yes
Alfil jump: no, yes
Dabbabah jump: no, yes
So 3*3*2*2*2=72 types of pieces total, including the "stone", which has no movement at all, but could be useful for hiding behind. Stones could also be usefully dropped in a crazyhouse variant.
Compute some endgame tablebases with these pieces. The royal piece can be selected from any of them, or a subset which all need to be captured. The interesting ones could be long forced wins. What others? By looking at draws, one can approximate piece values.