How do gravitational waves agree with Lorentz invariance?

Following is a simple but incorrect explanation for gravitational waves. My question is what is wrong with it?

I’d like to say that a gravitational wave is a periodic variation in the local gravitational field. For example, suppose the Earth is not rotating, for simplicity, and that the Moon orbits the Earth every 28 days. In this case, an observer on Earth would observe the Moon’s gravitational field changing with a 28 day period, which seems to me would be an observation of a gravitational wave. The observer could also go sit on Pluto and measure the local gravitational field there changing from Earth’s Moon. Again, this person would see it varying with a period of 28 days, but now delayed by about 5 hours due to the gravity transit time from Earth to Pluto. Again, this seems to me like an observation of a gravitational wave, but from a little farther away.

A problem with this explanation can be seen with the Earth measurement. From this explanation, I would expect the wave phase at Earth to be delayed by about 1 second from the Moon’s position due to the fact that it takes light (and gravity) about 1 second to get from the Moon to the Earth. This seems reasonable on the surface, but it violates Lorentz invariance, which in this case states that the gravitational field direction for an object moving at constant velocity should point directly toward the object (see Wikipedia “Speed of gravity”). The same issue applies for the Pluto measurement, too. Intuitively, it seems hard to believe that there isn’t a delay between the Moon’s gravity and its measurement on Pluto, but that’s what Lorentz invariance says. Admittedly, the Moon is accelerating very slowly, but that was not a central part of my explanation.

So, is my explanation some sort of “near-field” effect, and distinct from actual gravitational waves? Or am I missing something else?

Thanks for any replies.

-Steve