At high enough N, ZFC is independent of BB(N), and in fact any math axiom system has such an N.
The LLM itself is finite, the axioms it knows are fixed, there is an N where BB(N) is independent of those axioms, so the LLM cannot solve it.
The LLM itself is finite, the axioms it knows are fixed, there is an N where BB(N) is independent of those axioms, so the LLM cannot solve it.
To put another way, if it were true that some fixed LLM could solve every math problem, it would implement Halt, which is impossible.