So did the car allow the FOB to be programmed/re-programmed with the 10/10/10 method or when you say "nothing" do you mean the car didn't honk at all as if it's not recognizing that an FOB programming attempt is being made?
If it's the latter, even though you went through the massive pain(I tried and gave up with my old FOB) of replacing the battery in the old FOB, the chip board itself might have finally given up the ghost as it happened to me just a month ago. TL-DR bought a new Holden FOB from Juniorss on this forum and programmed it with the 10/10/10 and good as new.
Complete NEW OEM 2004 2005 2006 Pontiac GTO Key Fob Remote (Monaro logo)/ Key | eBay
That's the link I used off of eBay for him but it looks like he's sold through that batch of keys he had. Maybe message him on here directly and see if he has any available if the FOB replacement ends up being the solution to your issue.
EDIT #1
Up for sale is an OEM GM key fob for the 2004 2005 2006 Pontiac GTO. As you can see, the fob has the Holden logo on the back as opposed to the Pontiac logo. The Pontiac key fobs are no longer produced, but these are identical to the GTO fob except for the logo.
www.ebay.com
Here's a link for Junior's other eBay posting I just found for just the FOB without the blank spare key and without the two screws.
Best of luck mang.