Of course, Joseph is wrong halachically. There’s no reason the merchant needs to assume that the transaction is without the explicit or implicit permission of the husband. If he would want to challenge the transaction, he would need to prove that it was outside of their normal spending patterns and was without permission, which is nearly impossible to do.
This is is all assuming she didn’t say איני נזונת ואיני עושה, in which case the whole thing doesn’t start.