Knee, I would suggest two tourneys, concurrent or consecutive, however you want it. One for AL and one for NL. Every franchise would be represented and the teams would have to have been either in last place or have finished with a typically last-place record. I would put the high range at .420, which would be about a 68-94 record, or if the team finished last, the record wouldn't matter. (No pun intended on "high" and "420," by the way.) Whatever years you want to make it wouldn't matter to me, but I would suggest 1969 to present, which is the divisional era. Brewers can be represented in both leagues (1969-1997 in AL, including the 1969 Pilots, and 1998-2009 in the NL), and one franchise can be represented twice in the AL (must be different years) to make it 16 teams there.
Even better, the winners of the two tourneys can meet in a best-of-seven Losers' World Series.