Are there special requirements for the characters to bet their items? I have yet to get one.
You do not need to do anything special in order for the characters to bet their special items. It is simply a random occurrence. When you start up a new game, instead of doing the $10,000 buy in, the character will offer their special item. So you could theoretically just keep trying to start games and quit.
Okay, just lost about $40k testing this, but I have a definitive answer now.
In order to restart the game several times without losing $10k each time, make sure you hit Escape to bring up the menu before the hand is actually dealt. If you do this while the opening cinematic about the buy-ins is still going (everyone tossing their cash in) then you will not lose $10k. As soon as you see 4 bundles of $10k hit the table (which means no special item), hit Escape quickly or else the hand will automatically be dealt and the game will save that you've lost $10k. If the camera pans to someone during the buy-in cinematic, that means that they don't have the money and will add their special item that you can get.