They originally asked Ronald Reagan to be the town mayor but he thought it best not to. Perhaps they should have asked Huey Lewis instead. Oh well

That was a good version! I don't mind that they used ZZ Top, but they probably should have worked in another Huey cameo. Sure, they used "back in time" for the BTTF III trailer and you hear the song on the truck radio at the end with Needles, but Huey Lewis and the news became such a major part of the series and that era. I'm sure they could have come up with something.
You'll kill everyone!
But Ice Cream Cake!

It was indeed "Doubleback," although an acoustic instrumental version of it performed by the members of ZZ Top making a cameo appearance as the musicians at the town festival.

I think the song you're referring to is called "Doubleback," written particularly for the movie.

It was called "The Battle Cry of Freedom". It was a song sung by the Union Soldiers after they won the American Civil War.