I think while Dante was investigating the Order he discovered, probably through Trish, what they were really doing along w/ Yamato. I mean you don't just crash a religious ceremony and kill the priest w/out a REALLY good reason. Plus, Trish gave his "holiness" Sparda's sword and they never even talk about that. Since the Order collects devil arms, I think Trish (Gloria) gave him the sword to win his trust.
Trish (Gloria) was in their inner circle so she knew about the research Agnus was doing; using demons from Hell for experimentation. Therefore, upon investigating further, she discovered that they were using Yamato to open the gate to Hell, which is one of it's functions.
This is all stuff we should have seen in the game. I mean they wanted to push Nero so badly but they didn't even show us how he got his arm!!!! 
They have to release extra missions or a SE version. PLEASE CPACOM!!!! I NEED MORE DANTE!!!!! [:'(]