Well, this is just a historical detail and not worth a huge debate, but I wanted to state what I think would be more correct.
Just convert to Hinduism then.
Um, how do I convert before it spreads to my lands? The first city never keeps the pagan temple.
With pagan cities in Gallia, Rome and so on, where Christianity spreads, pagan temples should be mostly abandoned and rarely find new uses as churches, imo. The pagan pantheons are just too different from Christianity. Currently, this doesn't happen:
Baal temples of all places, are converted to
orthodox churches, if orthodoxy is state religion. It may be convenient, but it's not correct.
Hinduism, on the other hand, is based on vedic and brahmanic texts, that were only subsumed under the name 'hindu' in modern times. Vedic and brahmanic temples (proto-hinduism in the sense of our game) should therefore rather convert to Hinduistic temples once the 'official' religion spreads there, regardless of state religion. And they should coexist with Buddhism, or be abandoned when any other religion spreads.
Just like Olympian (pagan) temples coexist with Judaism and you can even built new ones (!) if Judaism is present. Did that as Greece recently, and had an olympian temple in a jewish city, which I thought was weird. Especially when the temple became an orthodox church later.
On the other hand again, I have seen churches disappear in thin air when another christian confession spreads. In my opinion, christian churches should be kept under the new confession (repurposed churches are common enough).
I have to check about the inquisition unit, but I'm sure there is also no conversion happening. Example case: state catholicism doesn't have a church but is present in city, Inquisitor purges, say, orthodoxy, which has a church, monastery and cathedral. I would expect the monastery to be dissolved (destroyed for gold), but church and cathedral are repurposed as catholic. Of course, this mechanic wouldn't apply to non-christian faiths, where all places of worship would be destroyed for gold.