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Your comments about the Alfil and Dabbabah remind me of the Dragon in
British Chess. This piece is a compound Alfilrider and Dabbabahrider. So,
like the Dabbabah, it is limited to only one quarter of the board. Each
player gets two Dragons, which are enough to cover only half the board,
and the four initial Dragons in the game each cover a different quarter of
the board. The only way for two Dragons to cover the same area would be
through Pawn promotion to a Dragon. But since the only way a Pawn may
promote to a Dragon is if one has been captured, no player will ever have
more than two Dragons.

Despite the fact that a player will never be able to cover the whole board
with his Dragons, I don't think the game suffers from giving each player
only two Dragons instead of four. The Dragon is useful mainly in support
of other pieces. Also, given that a player's Dragons cannot capture each
other, there is a greater potential for uneven piece exchanges, which may
help to make the game more interesting.