That is not Weird Al. There are 100s of songs credited to Weird Al that are not his.
Ok. Well this one is definitely his, and fits into a wedding setting 
It does. See post # 46.
He Hit Me (And It Felt Like A Kiss) by The Crystals.
Such a beautiful song celebrating domestic violence and a woman who recognizes her place after possibly cheating or looking at another guy the wrong way, saying that being hit feels like a kiss, and she likes it.
Best wedding song ever!
And if you’re going with Weird Al, this one is very good one.
I came into say “Bad Girlfriend” by Theory of a deadman, but someone beat me to it! Great song, and the singer actually did write it for his wife.
What If I’m Right by Sandi Thom
If I Didn’t Have You (Someone Else Would Do) by Tim Minchin
How about
50 Ways to leave your lover.
When I’m 64
The Bitch is Back
[ul]
[li]Grey DeLisle—“This White Circle” (On my finger means we’re through)[/li][li]George Jones—If Drinkin Don’t Kill Me (Her Memory Will)[/li][li]Patsy Cline—Why Can’t He Be You[/li][/ul]
I got the bride’s permission to use this one. I’m going to introduce it with something like “Scott just handed me this song he wrote and performed…”
ETA - Weird Al’s “Good Enough for Now.”
I talked it over with the bride’s father. He thought it would be hilarious. I might use it in a “I don’t know much about country music so I just downloaded a few songs from a list of top country songs I found on the internet” and edit it so that it starts with the chorus, play a few lines (get laughs hopefully) and then stop. Then maybe move to the chorus of D-I-V-O-R-C-E.
Fifty Ways to Leave Your Lover, Paul Simon.
The Offspring’s “Why Don’t You Get A Job?” could be funny., crowd-dependent.
Highway to Hell seems in appropriate for newlyweds, but suitably funny.
She’s So Mean by Matchbox Twenty
I win:
Third rate romance – The Amazing Rhythm Aces
I Hate My Life is another song by Theory of a Deadman that might fit.
But I really came here to suggest If You Leave Me, Can I Come Too? by Mental As Anything. Catchy melody and danceability.
The Stones’Under My Thumb
The Buzzocks “Ever Fallen in Love (with Someone You Shouldn’t Have Fallen in Love With?)”
The Smothers Brothers “I Never Will Marry”