Not all of these are suitable to share (and whether you share any at all is up to you). But maybe some will help you. Some aren’t exactly about breaking up, more about loss, but I think they kind of fit.
Time to Move On - Tom Petty
Wildflowers - “”
Another two off a single album:
Walk On - U2
Kite - U2 (maybe the closest to what you asked for)
April Come She Will - Simon & Garfunkel
Fare Thee Well - Indigo Girls
Taste the Pain - Red Hot Chili Peppers (this may not fit because it’s somewhat mean but not bitter)
Love Love Love - The Mountain Goats (this isn’t a breakup song at all and yet it somehow it feels like one to me)
Time of Your Life (Good Riddance) - Green Day (obvious choice, though different people see it differently)
Only The Young - Brandon Flowers
Don’t Think Twice, It’s Alright - Bob Dylan (kinda iffy how bitter this is, but sometimes it’s just perfect)
I’ll Remember You - Bob Dylan (a bit nicer)
Walkaways - Counting Crows (short and to the point)
Dreaming My Dreams With You - Waylon Jennings (I like the Cowboy Junkies cover)
Tear-Stained Letter - Johnny Cash (the lyrics are downbeat, but the music goes against that)
At This Moment - Billy & The Beaters (those of a certain age will probably just find this amusing)
The Pain is Gone - Black Lab
Train Wreck - Glen Philips
Now I Understand - Lisa Loeb
These White Lights Will Bend to Make Blue - Azure Ray