I was curious about the 370 years... but wasn't prepared to crawl down that rabbit-hole until I understood how the context works. Eg if there was a special date you were working backwards to.
I figure that's enough time to pretty much own every cardinal if you're an immortal. You do have a limitation that the plants become bold over time and make mistakes, but through 4 centuries, I bet you get that 1 in 100 who's too crafty to fall that way. After I got some of those guys, I'd grant them immortality - I'd only need about 3 in the whole organization.
My evil plan would be:
a) Keep making stupid minions whose sole job is to get caught so I can monitor how good the church is getting at finding me
b) I'd introduce translation errors in some of the popular bibles such as NIV. I'd sow confusion by painting the devil's number as 665 and 667... I'd make sure dead sea scroll show up and split the faithful over verbiage
c) I would NOT try to get my evil guy into the seat of the pope. I would work hard to break in a good pope who I hold on a leash (Think Wellington Yueh). This is the best guy cause he can pass any radars, right? Everyone trusts him. Once he's in place, he doesn't need to last long... 8 months should do. I'll find sneaky ways to break the succession so the next guy they choose is also tainted. Give me 370 years, I bet I could do it.
I think 1876 would have been good. Hit the cardinals I don't control with the plague right before a big vote... sneak my suk-doctor-pope in. Then off him and replace him with my real guy. Then use my real guy to find and destroy some of the idiots working for me who I don't like, thereby validating he can root out agents of the beast.
"90 years": That's only ten popes or so. Might be harder for my evil plan to get a guy in the key spot in this time. I'd have done it during WWI then use the 1918 H1N1 to ratify guy #2.
"25 Years": Virtually impossible
You can see the pattern in all these ideas. Sneak someone in, and tilt the scales. Then feed him to the machine and back off. Wait till everyone dies off and pull the trigger.
Question: How relevant is the time scale? What are the shortest and longest time you've looked at?