This situation is very familiar to me.
When I decided to start focusing on playing contrabassoon, I started off by borrowing an instrument from the local university. On that instrument, not only could I not get the half-hole G to attack reliably, but I couldn't slur to it from the octave below at all. The latter was especially frustrating at the time because I had an orchestral part that asked for it in an exposed passage. (I don't remember now what the piece was.) For a while, I was very frustrated with myself.
I took a lesson with a prominent contra bassoonist. He let me try his Heckel. The very first thing I played was the slur, followed by the half-hole G. They were smooth as butter.
That's when I learned that the issue was the instrument, not me. That lesson, and that experience, was a turning point for me.
My long-term solution was to order and purchase my instrument. That does not help you with your Amati.
Here are a few thoughts for you and that instrument.
Definitely try different bocals (Heckel, etc.). Ditto different reeds and reed shapes (K1, K2, Fox standard, Fox Fast, Fox Fast 2, Burl Lane, etc.) and scrapes.
Try playing the G using the lower vent as a flick key (this may help clarify the attack and minimize the tone quality issue).
I like Ian White's suggestion.
Here is another thought. When I took delivery of my Fox, I noticed that the half-hole G didn't speak as cleanly as I wanted. Chip Owen opened up one of the holes in the wing joint a little bit. This could possibly be done with your Amati by a trusted repairman who knows what he or she is doing.
One more thought: the Fox Divorced Low E mechanism helps address the tone color issue when using the lower vent key. I do not know how practical or expensive it would be to retrofit it to your Amati. If you're willing to do so, you may find that it helps improve that G when using the lower vent key, as well as other notes (especially the Ab above it).