The 10 year old is probably risk averse on this, so would have good reason not to take such a bet--pretty much anything that he would lose for losing the bet is a loss that he wouldn't be able to absorb, and no sane parent is going to give him a punishment that is small enough that he can absorb it. Of course, being a 10 year old, he wouldn't use the term "risk-averse" and may phrase it in an awkward way or be unable to put his objection into coherent words at all, but that's basically what he'd be doing.

So the fact that he would likely refuse such a bet (unless he is overly optimistic, also a possibility) won't prove that he is wrong. As a parent, you could always lie and say "if you won't take the bet, that means you don't really believe it" but you would be lying, so whether you should do that depends on what you think about using bad reasoning to make kids obey.

On the other hand, it would be perfectly rationalist to say "I'm a grownup. I know how people, and especially kids, act. If I let you swear at home you will say it somewhere else, so no, I'm not going to let you swear at home." You're not criticizing his reasoning, you're asserting that you have better priors than he does, and you most likely do.