I would agree with @Wildan a player can use other computer to cheat so it doesn't make sense on having a bet on a Chess game but that would be great if anyone can stop people from cheating (Which is not) for me adding a bet on Chess game could spice up or make the game more intense.
It is possible to have a this kind of game if the player could have a live webcam to them (to see if they are using other computer) and a screen share of their computer just like what they are doing in online chess tournament but that was too much right? Setting up cameras and sharing your screen to others.
Yes that could, many monitors already have a camera inside, than you just need 1-2 more to film the PC and keyboard from behind. It's a bit like becoming a professional steamer. For some players this could be a no go, like you can't do on your phone while lying on the couch or at night in bed. Cheating would still be an issue though if stakes are too high. You could have a partner for example who watches the game life and feeds it into a chess computer, giving real time information directly into the earphone of the player. I think most important would be to focus on blitz games, there the games are much faster and less time to manipulate. It will also take a anti cheat software that compares your rank to the way you play. Like if you are winning too many games, or make some very good moves which would indicate a higher rating.