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DAN+, Different Kings and Pawns?
Changing the move of the King or the Pawns is dangerous.
For example, if one side has a normal King, and the other side has a
King that jumps one or two squares Rookwise, the side with the
normal King will win every King-and-Pawn endgame.
Another example is Berolina Chess, in which Pawns move diagonally
but capture Rookwise (just the reverse of normal Pawns); the opening
move for White of 1. a2-c4 or 1. h2-f4 is strongly indicated
if White has normal Rooks!
Changing the Pawns would prevent the creation of locked Pawn chains,
which is important to the look and feel of Chess.
All in all, I cannot currently recommend substitutes for the
standard King and pawns.