I originally chose 370 back when I was still crucifying cardinals every 5-10 years. The A-C had to move slowly, otherwise the Church would have been forced to mount an aggressive defense, which would have spilled into the open. When I switched the A-C's approach to infiltrating at multiple levels, I simply kept 370 as a placeholder until I figured out if it needed adjusting. It does.
Things to consider, though. The Church regularly purges certain offices (just to be unpredictable and to clean house). All the key positions are given to cardinals, who are usually old men with a very limited remaining lifespan. That prevents anyone staying in place too long. The estimated number of infiltrators is 10K. There are 5600 bishops in the world. If the A-C wants a near-perpetual lock on the key dioceses, he has to have obvious replacements in line for those dioceses. He also has to have control of key Vatican dicasteries (departments), plus a few in line to succeed those who head those dicasteries. So, 10K stretches the A-C kinda thin, especially when you consider that a decent number of his followers will be discovered because they blow their covers with bad behavior.
Related to all this is the fact that the A-C has to have his followers at or near their target positions when he takes control of the Church. There's only so long he can stay in power before people realize he's not purely human due to the fact that he doesn't die of old age. He's also racing to beat the clock, because when Jesus returns, times up. If Satan's challenge succeeds, Jesus will be destroyed upon his return and the Church will crumble thanks to the A-C and his followers.
There are actually three members of the Unholy Trinity, one of whom is the False Prophet. In the early Church, it was generally believed that the A-C was a secular ruler (e.g., Emperor Nero) and the False Prophet was a holy man. That made the latter more suited to seizing the Church. However, protestants confused all that by equating the Catholic Pope with the A-C. In modern times, there is no global position of secular power at risk of being seized by the A-C. He would need to engineer control of the US, Russia, and China in order to have a real shot at world domination. So, when choosing which interpretation of Revelation to go with (there are many), I decided the A-C will go after the Church. The False Prophet plays a lesser role, and I punted him to book two. He's a key character in book one, but remains under deep cover, working mischief.
Off to bed.