You can't use it to build a self-contradicting Turing Machine, because a hypercomputer cannot be emulated by a Turing Machine.

There is no "the" halting problem. Each class of machines has its own halting problem. The halting problem for turing machines is very hard. The halting problem for non-cyclic finite state automatons is very easy. A hypercomputer that can solve all halting problems is self-contradictory. A hypercomputer that can solve turing-or-weaker halting problems is not self-contradictory.

In this kind of context, you should provide a link. This kind of thing can spill over into "You're wrong because you haven't done the work to see that I'm right", and minimizing the work you're pushing off shows good faith as well as increasing clarity that we're talking about the same thing.

I think it's more likely we're talking about different things than either of us failing to understand the relevant proofs. I'll take a look, though, when you've provided the link (and ideally some explanation of what you see as important).