On the classical side, I second the suggestion for Salut d'Amour and Chanson de matin.
On the folky side, I think you'd want to keep away from true slow airs - they're beautiful but somewhat depressing (!) and because they aren't metered, they can be strange to listen to if you're not used to them. What about transcribing some folk songs and making them into tunes? She moved through the fair would be nice, as would skibbereen and Erin gra mo chroi.
You could also play a slow strathspey (captain campbell is fairly easy) or take some of the prettier jigs and reels and slow them right down - crabs in the skillet would work nicely for a jig and you could check out the farewell to midtown for a nice reel that works slowly (both in g minor).
Another possibility would be some airs (rather than slow airs) - the scottish fiddle music is by far the richest here - things like Calum sgaire or Neil Gow's lament for the death of his second wife (tell people it's called the fairy bluebell and they all lived happily ever after best wedding ever song if they ask
) - I'd also suggest the nameless lassie, hector the hero and will ye no come back again. All beautiful.
Hope it goes well!
Thanks, Rosfrog! I'm going to tentatively suggest replacing the wedding march with something else, and float the idea of something folky and see how it goes down.