Had a similar issue with an A string on non-Modulus bass, turned out the slot for the A string on the nut was too wide so it would rattle when played open. I would check to see if the slot for the G is worn on the lateral edges or seems wider than the string. Aside from building up the edges with a mixture of cyanoacrylate and baking soda (if it's white plastic or bone - you can get black cyanoacrylate as well) and then reshaping the slot, you could try going up to a slightly thicker gauge string. A quick test would be to insert a thin piece of folded aluminum foil under the string in the slot, just enough to ensure that string sits snuggle in the slot. If it stops rattling as you describe, then that could be the culprit.
Another thing I do to avoid issues at the bridge is to press down on each string just after the saddle to create a crease in the string or witness point when putting a new set of strings on - also helps with intonation.
And, it could simply be that the G string is not high enough to allow for wider vibrations when you're digging in. Try raising the saddle slightly.