Buzzing on the first few frets is a classic indication of not enough relief.
It is sort of strange that it only buzzes on the G string though. Usually the larger diameter strings like B and E are the last ones to stop buzzing as relief is added to the neck.
My suggestion is to go another 1/8 of a turn on the TR. You can always go back the other way if needed and loosening the TR nut isn't likely to cause any damage.
After reading about the TR breaking on the other thread concerning a Corvette, I'm almost afraid to offer any kind of advice.
Be very careful that the wrench fits the nut properly. You sure don't want to round out the socket that the allen key fits in.
Pkr2