Note that the halting problem isn't very relevant here. You can take a much simpler problem, like computing the sum of two integers. By the same argument, it's just as impossible to fully test a black box that claims to be an adding machine, but outputs garbage for inputs greater than some N. Moreover, you can't always determine whether a box is an adding machine even if you're allowed to look inside the box and inspect its algorithm, by Rice's theorem.
Here's something I've been wondering about, in the context of Solomonoff induction and uncomputable sequences.
I have a device that is either a halting oracle, or an ordinary Turing machine which gives the correct answer to the halting problem for all programs smaller than some finite length N but always outputs "does not halt" when asked to evaluate programs larger than N. If you don't know what N is and you don't have infinite time, is there a way to tell the difference between the actual halting oracle (which gives correct answers for all possible programs) and a "fake" halting oracle which starts giving wrong answers for some N that just happens to be larger than any program that you've tested so far?
The Kolmogorov complexity of an uncomputable sequence is infinite, so Solomonoff induction assigns it a probability of zero, but there's always a computable number with less than epsilon error, so would this ever actually matter?