You want to know whether the following is a threat to Anselm’s ontological argument:

1. The only way to prove something a priori is if its opposite is a contradiction.

2. If something is a contradiction, it is inconceivable.

3. Everything can be conceived not to exist.

4. Therefore, nothing can be proven to exist a priori, including God.

You’ve read, or perhaps listened to, my lecture on Anselm, so you know that the argument doesn’t actually turn on the claim that the nonexistence of God is contradictory and that the existence of God is therefore necessary, though in fact one comes to acknowledge this necessity as result of seeing in Proslogion 2 that “that than which nothing greater can be conceived” cannot exist in the mind alone.

Much in this Humean criticism turns on the word “conceive”: Hume is certainly right that if something is contradictory it can’t really be conceived. But it’s nonetheless easy enough to imagine we’re “conceiving” something just because it’s expressed as a noun or nominal phrase in a syntactically sound English sentence. It’s not obvious to most people at first glance, for example, that “a rock too heavy for God to lift” is a contradiction, and so they think they’re “conceiving” such a rock as a real possibility. This is the atheist’s mistake with regard to the supposed non-existence of God.

It’s also important to keep in mind that for Hume only those things that disclose themselves in facts of an empirical kind can be truly said to exist, and of course every such thing, being contingent, can be conceived not to exist. But here’s where the third of Saint Thomas’s ways kicks in: if everything were contingent, then there could have been some time, or some possible state of affairs, in which nothing existed. But ex nihilo nihil fit. Or alternatively one could perhaps borrow from Descartes and point out that the “conceiver” who conceives of “everything” as not existing cannot conceive of himself not existing; hence he cannot really conceive of everything as not existing.

These are the lines along which I would direct my thoughts if I were you.