Since large cardinal axioms have arithmetic consequences not provable without them, the answer to your first question is yes. For example, let $f(n)=1$, provided that $n$ is not the Goedel code of a proof of a contradiction in ZFC. Since large cardinals imply Con(ZFC), the theory ZFC+large cardinals proves that $f$ is total. But ZFC alone does not prove this (if consistent), since it is relatively consistent with ZFC that there are proofs of contradictions from ZFC.
For the more general question, suppose that the stronger theory not only proves the weaker theory, but also proves that whenever the weaker theory proves that a TM program computes a total function, then it really does. This is the situation for most large cardinals---for example the theory asserting that there is a weakly compact cardinal implies towers of smaller inaccessible cardinals, and so if a program is total inside such a $V_\kappa$, then it really is in $V$. Given this situation, let $f(n)$ be the function which first inspects all smaller $m\leq n$ to see which code proofs from the weaker theory that a certain function $g_m$ is total, and then let $f(n)$ be larger than all such $g_m(n)$. Our assumption on the stronger theory ensures that we can be confident that the computation of $g_m(n)$ converges, so $f(n)$ will be defined.