Well, since you specifically requested non-folk type songs, I'm not going to suggest any folk-type songs. :-)

"My Only Love" was a hit for the Statler Brothers on the country charts in 1984(?). It's an actual wedding song (not just a love song), and it's not "too country."

"One Hand, One Heart" goes back more than 40 years, but being from "West Side Story" it's likely somewhat familiar to your audience. Again, it's an actual wedding song. I find it to be lovely and elegantly simple.

We had these songs as part of our wedding, and it's been 22 years now, so they must work! :-)