Consider the problem of taking an input Turing machine and determining if the final cell is a $0$ or $1$ after computation halts. On cases where it writes something else or does not halt, you are allowed to give any answer (but you have to halt and give some answer on all inputs).

Is this problem undecidable? My gut says that it should be, but I can't find a reduction to the halting problem. Given a Turing machine that may or may not halt, we can set up the machine to finish with a $0$ in the case that it halts, but can't finish with anything in the non-halting case, so the oracle could just say $0$ in this case without having to figure out whether in fact the machine halts.

Note that a reduction in the other direction is simple; if you can solve the halting problem, then given a TM that either finishes with $0$ or $1$, we replace the $1$-writing step with an infinite loop to create a new TM. If the new TM halts, we say "it writes a $0$" and if it does not halt we say "it writes a $1$". This answer is guaranteed to be correct as long as the TM in fact halts with a $0$ or $1$, so we can solve the original problem.