I'm not sure that my paradox even requires the proof system to prove it's own consistency.

If the system is inconsistent, your program will halt on all inputs with output dependent on whether it happens to find a proof of "this program halts" or "this program doesn't halt" first.

Of course a crippled logic can't prove interesting things about Turing machines.

Well, mathematicians have been proving interesting things about Turing machines for the past century despite these limitations.