Its also worth noting that you can buy carbide tipped lathe tools, I have quite a few at home for hard steels. You can also buy carbide inserts as well. I only mention this because cutting carbide rod down to size might be problematic.
Overclocked, you beat me to it! The carbide rod basically can only be cut with diamond tools. But the carbide inserts would be what I would use. They are very close size wise, one to the next, and would not need further machining. Some times you can even get used ones from a machine shop for free or at least cheaper than new ones. In machining the corner/edge is what wears/breaks, the thickness stays the same. And thickness is what would be important in a spark gap.