I have not visited this gambling website till now. But I don't think there is any need to stop auto bet when switching to other tabs. Users make auto bets on their own will and without any force. You can also add stop loss to stop the auto bet once the balance becomes lower than a specific amount.
You made a great suggestion, that is as same as stop-limit orders for trading. Gamblers should determine which amount of money they want to spend on gambling in general and for auto-bet in particular at the start. They should determine which amount of money they might accept to lose with auto-bet before using it.
From my perspective, if someone uses auto-bet, the low odds should be chosen and set for the auto-bet bot in order to prevent serious losses in a short period of time.