This is neat. I’ve never heard of moustache distortion before!
Yeah, I’d be curious what the automatic lens correction does. When I downloaded your photo, opened it in Photoshop and ran the Filter > Lens Correction tool on it, a lens profile was automatically applied and seemed to fix the problem properly (see first attachment).
When I tried to fix the distortion manually, I started having the same problem as you described (see second attachment)! So yes, fixing it manually does not seem to be the ideal way to do it.
If, for whatever reason, you do want to fix it manually, then two ideas came to mind. #1, you could clone from areas on the horizon that are straight. And #2, you could take it into Photoshop and use the Edit > Free Transform tool to make the horizon do whatever you want (see third attachment)!