The song that comes to mind is "The Next Ten Minutes" from The Last Five Years. In the context of the show, the couple does break up, but as a stand-alone song, it's absolutely beautiful.
I think I remember Santino Fontana saying that Victoria Clark sang "They Were You" from The Fantasticks at his wedding.
A song that I've always wanted played at my future wedding is "The Human Heart" from Once On This Island. I will love that song until the day I die. Hope this helps!
Also, for the sake of 2020, just do "Wartime Wedding" from Over Here!
"See a picture of a woman wearing four years of confusion like a scar."