No, it has been proven that there exists no turing machine that can solve the halting problem. If we assume that there exists a computable f(N) that is always greater than BB(N) then we can construct a Turing machine that solves the halting problem. Therefore no computable f(N) can exist.
I think the issue is related to notion of a "direct" proof vs "proof by contradiction". What I gave was a proof by contradiction. Once you start thinking about the philosophy of proofs it, it's an interesting question to ask whether or not proofs by contradiction should really count. I don't know much about this but you can search up the term "Law of the excluded middle" is a good place to start reading about these concepts.
Trust me, I understand proof by contradiction :). The point I was missing was the difference between an undecidable question vs impossible. See the rest of the thread above.
