I don’t recall this one at all. The Huey Lewis video was in heavy rotation at the time. That looks like it aired on one of the music video shows that competed with MTV before exclusivity deals killed them off.
I’m thinking it’s from “Friday Night Videos” which I think aired on NBC back when Letterman was only on 4 days a week.
I’m surprised none of that extra footage (especially the end) was ever in any of the BTTF documentaries, although I haven’t seen the new ones on the 2015 blu-ray release.