I would prefer more than three players for the Chess-Xiangqi-Shogi phase.
 
How about a Shogi event starting in January, then a Chess event starting
in March, then a Xiangqi event starting in May (because Shogi tends to
take more time than Chess, which tends to take more time than Xiangqi)?.
Each event would have the players moving twice as White and twice as
Black.  If say three players tie for first, the BS/SB method ranks them
for tie-breaks, with player 2 moving first against player 3, and the
winner moving second against player 1, so that there are seldom more than
six games for an individual event (and only four if you don't tie for
first).

Then you give 1 point to the winner, 0.5 point to the runner up, 0.333 to
the third ranked player, and so on... in each game, and you qualify the
seven better players for the second round (thus, the winners for a single
event always get qualified). These seven players move each once as White
and once as Black at each game, with a higher BS/SB ranking in a tie-break
situation giving the higher-ranked player the right either to move first
(and the lower-ranked player choosing at which game) or to choose between
Chess, Xiangqi or Shogi (and the lower-ranked player moving first).