@raville been thinking about your proposal, it doesn't actually prevent playing always the same side. Assume a player always playing and winning with the Allies, he loses the Axis bracket, then wins the Final with the Allies again.
If we really want to force a player to change side along the tournament, we could introduce an "Allies Index" attached to each player.
Each player starts the tournament with "Allies Index" = 0.
Players get +1 to their Index for each game won on the Allies side and -1 to their Index for each game won on the Axis side.
For each new game, the bid is modified by adding to it the "Allies Index" of the opponent and subtracting to it the player's own "Allies Index".
Let's assume in the third round of the tourney two players face each other, one is very good with Allies, he won both his previous games with the Allies and therefore his Index = 2, the second player is very good with the Axis, his Index = -2.
Now they must be very careful with their bidding process, as the Allies lover gets a -4 modifier if his bid is lower, while the Axis lover gets a +4 modifier for "accepting" the Allies.
Now the effect of introducing this index is very interesting and probably goes beyond that of forcing players to swap sides during the tournament.
By winning with the Axis, in fact, you are actually building a small bonus to play Allies next. This will most likely lead to an average increase of the bids to play the Allies, which will on turn give way to some unprecedented scenarios