Answer

The Talmud says that vows are considered something that has a way to become permitted, because they can be dissolved through petition. We see from there that even if the permission involves an act, it is still considered something that has a way to become permitted. True, there the act is going and asking for dissolution, whereas here we are talking about giving money. As I understand it, this is not called something that has a way to become permitted, since he has to take money out of his pocket in order to permit it. The logic of something that has a way to become permitted is that one should not be allowed to eat it while prohibited if he can eat it permissibly. But if he has to spend money for that, it is not considered that he can eat it permissibly.
And in a somewhat different formulation: the whole idea of nullification by majority is saving money. The Torah does not require you to forgo permitted food merely because of a concern for prohibition, unless the prohibited part outweighs the permitted part. But if it has a way to become permitted, then he loses nothing, and there is no reason to permit it to him. If, however, he must spend money in order to eat it, then he is in fact losing money, and so it is not reasonable to say that he would be forbidden in such a case.
However, here we are dealing with a vow prohibition, meaning he vowed not to benefit from the item, and vow prohibitions in general are considered something that has a way to become permitted. So the bottom line is that such an item does have a way to become permitted—not because he can give charity, but because he can seek dissolution of the vow.