Undecidability of two Turing machines acting the same way on an input
You were very close to the answer. The special case you mentioned can be handled by hand. Modify the machines resulting from the reduction such that, for each transition, where the machine halts and rejects, let the resulting machine go into an endless loop instead.

Now each machine either accepts or goes into an endless loop. That means, checking if both machines are equivalent on an input $x$ is checking whether both accept the input $x$.