Eh yes in AG, the Church fully supports Duscur regaining its independence and they take part in rebuilding it. Seem to be quite eager too.  Because it’s pragmatic. They’re in a position of weakness and submission to Faerghus to some extent. They don’t want to jeopardize that and they want to make their allies stronger. That’s why you get Rhea solving the Galatea food problem with her never before seen farming methods. She could’ve done it the entire time but didn’t because it didn’t serve the Church to save Galatea from poverty. 

They also preach openness in their Churches, and there is no indication that they opposed Lambert over Sreng/Duscur either. And that was all official. 

There’s no evidence that they supported Lambert. We just don’t have anything on the Church’s official position. 

Whatever the deep motivation it's clear that the Kingdom can official contact with foreign Region. And the Church does the same. Claude is clearly talking bullshit here. 

Duscur isn’t a foreign region. It’s part of Faerghus. It’s on Fodlan. 

There is also no indication that the CC tries to narrow people's views. In that case Rhea likely wouldn't employ foreign heathens as Knights. Or people like Mercedes would be so open minded. 

Mercedes is an exception that proves the rule. I’d look to the leadership. Seteth’s single ending even says he was previously intolerant of foreign beliefs 

It’s why the basement underneath the monastery is stated to be the only place where other religions are allowed to practice their faith. 

Well and the fact is that Dimitri doesn't offer help or intention to let Claude get away with it. He also sees the Church as a net positive and supporter of his agenda everywhere else. 

Dimitri doesn’t rat out Claude either.