It's not really the church as a whole, but only two of their cardinals, if I recall correctly. The motive? Personal profit, I think? Or the increase in power of their church. Don't remember.
The important part of that the Lord they want to marry Nell to, is the one orchestrating everything. He's the military's head honcho.
He is extremely patriotic and his plan is to take all the miscreant nobles in his nation into "his faction" (which they actually made up on their own without noticing that Military Man didn't actually participate nor verbally agree to any of their plans) to make them expose themselves for easier elimination.
He already figured out that Nell's fiancé is the Demon Lord of the forest on their borders. When Yuki uncovers his involvement in Bad Guys Inc., he strikes a deal with Yuki to look after the nation in exchange for him protecting Nell from internal bad actors.
There are more details, but that's the jist of what I remember.
So yeah, not a typical idiot/malevolent noble plot. Though, not the first time I see it either.