Would that wave be that destructive? I can definitely see it screwing up the orbits of Jupiter’s moons, maybe even our own moon, but would it be much worse than a small earthquake?

The Sun’s gravity at Earth’s distance is only 0.0059m/s². I’m not exactly certain about how the magnetude of a gravity wave relates to the magnetude of the static force, but even if the force fluctuates rapidly at ten times the static force, that’s less than a hundredth of a g; enough to be perceivable but you wouldn’t even loose your balance.