I'm really curious to see how this plays out. I attempted this on my own with Isabella on Monarch over the bank holiday to see how well I would fare and compare it to your game. I have never built the AP yet, and I haven't had any losses to it either, so this was my first real experience with it. I hope you do better with the AP than I did.
I was on a continent with Hammurabi, Sury and Mansa. Jaoa and Ragnar were on another continent that could be reached through a couple of islands with fishing boats and galleys. Gandhi was on a continent by himself that was also able to reach Jaoa and Ragnar with fishing boats from the other side.
Gandhi got Buddhism and I got Hinduism, then Judaism and Christianity. Hindu spread to Hammy, Sury, Mansa and Ragnar. I built the AP and we were all buddy buddy. I spread Hinduism to one of Jaoa's cities and then finally to Gandhi with Optics.
Unfortunately Sury had just enough votes to foil my win. I figured I'd liberate a couple of Hindu cities from him to weaken his vote and strengthen mine. That worked well. By that time, I had no real competition but in spite of Hammy and Mansa being friendly and pleased with me, they abstained from the vote. I still fell shy.
I REXed like crazy and built a huge Hindu empire, but Jaoa kept creating colonies forcing me to go convert a city on some island to get the vote. I finally got everyone on earth to have some hindu influence and with relatively good relations with everyone but Sury, who nobody really liked anyway, everyone still abstained. I never broke the cusp of having enough votes on my own to win.
I did get the space race victory because I had planted cities everywhere I could to make them hindu and get votes. It was far more cities than I normally use, and I think I learned how effective that can be when building workshops everywhere. I was way ahead in techs and production and cruised to Alpha Centauri.
It was a strange game, because I've never waited so long to go to war and I was way behind in the power graph from my REXing and spamming. My neighbours liked me though, so I didn't get anyone declaring on me. Hammurabi snagged both Taoism and Islam, yet never spread them or converted.
In hindsight I can say that the best way to leverage the AP really is to hoard the religion yourself. Don't let it get more than a toehold in other civs, which is kind of counter intuitive. Hinduism was ultimately just too dominant in the world and that prevented my win.
By the way, is Hammurabi always such a weenie? For crying out loud, he's aggressive and I couldn't bribe him into war with anyone. He spent the whole game just peacefully building an army that sat there.