1. What tilt is undergoing a "wobble"? The Eart's axial tilt?
2. If so, exactly what is the magnitude of the wobble, and its period? Hint: according to the above quote, there's only one answer.
3. If the Earth is "wobbling" as you seem to propose, no fixed telescope on Earth would track properly.. Many, many such telescopes exist both in professional and amateur observatories. Yet no such problems have occurred.
4. Such a daily wobble would throw off every geosynchronous satellite, not to mention screw up the GPS system, etc. However, there have been no such problems.
5. Where is this "12th planet", exactly? What are its coordinates and visual magnitude? How large is it? What's its composition? What are its orbital elements?
6. You seem to be claiming the Earth translated significantly perpendicular to its orbital plane, or that the inclination of the orbit itself changed temporarily. Which is it?
6a. If you're claiming the former, you can't just say it moved up but stayed in the same orbital inclination. That's unphysical for a central force like gravity. So, either way, the orbital inclination of the Earth would have to change. This also would render every fixed telescope on Earth useless - unable to track any star, planet, or the Sun.
6b. We'd also immediately notice that the sunrise and sunset times, etc., were wrong.
6c. Either would be an awfully energetic event. Where did all that energy go?
7. If this "12th planet" is big enough and close enough to push/pull the Earth around in such a fashion, why is the Moon still in orbit? Why is Mars still in its orbit? Why did Cassini arrive exactly as targeted at Saturn, or Genesis make it back to Earth?

Your assertions that the Earth has tilted, or moved up or down out of its orbital plane, or that its orbital plane has shifted, are falsified by the lack of any of the above consequences being observed.