Dumb question: does using an unstable isotope introduce any wobble? Could this method work with a stable element?

What do you mean? The half-life is ~8000y so its not like you'll run out. Do you mean the isomeric state is different because the nucleus is unstable long-term, I'm not sure that this matters.

From the article: “Nuclear transitions, by contrast, usually occur at energies far above typical electronic transitions, leaving most nuclear resonances beyond direct coherent access with conventional optical spectroscopy. The 229Th isomer is a singular exception. Its transition energy lies in the vacuum-ultraviolet (VUV) range, making it the only known nuclear transition that can be addressed directly with current laser technology.”