I'm going to offer an alternative interpretation/explanation:
"May not" doesn't work because of two problems:
- Least important, you're mixing up present tense ("you may not exchange it") with an implied future tense. It's not terrible; everyone would be able to understand what you mean. But it definitely sounds off. "Can't" technically should suffer the same problem, in that case it would sound less off because "can't" is often used sloppily in casual speech where as "may" is not.
- "May not" refers specifically to permission, not possibility, and heavily implies that you yourself are imposing the restriction on the person you're talking to. If you are instead describing someone else's restriction, as in the case of a customer service representative telling a customer about the store's return policy, you would use the "allowed to" (e.g. "You're not allowed to eat here."), or say whether or not it's doable (with "can" or "able to", e.g. "You can't eat here.").
The meaning of "Otherwise, you might not exchange it" doesn't even make sense; it means, "Otherwise, there is a possibility that you will [of your own free will] not exchange it." It's redundant, because the person you're talking to already knows that they can make a choice.
In an attempt to make this more clear, "might" is essentially a prediction of a possible future event. It is
not a word that addresses whether or not something is allowed, nor is it a word that addresses whether or not something is doable. So for example, if you were to say, "I might eat that sandwich," that does not mean that you are able to eat the sandwich; rather, it means that you haven't decided yet whether or not you are going to eat the sandwich. Alternatively, if you were to say, "John might drive drunk tonight," it doesn't mean that John is allowed to drive drunk, rather that you expect that to be a realistic possibility (perhaps because you know he drinks alcohol irresponsibly).