How to add a TileSet (or TileMap) background?

You need a tileset for the background? I think you can just set the z_index of the tileset to a negative number, and then it should be drawn behind all of the other Node2D-based nodes, while still functioning like a normal tileset. You may also need to disable collision though, unless you want your entities to collide with the background, but there is probably a way to do that as well.

I wonder. Might the parallax background be of interest perhaps?

https://docs.godotengine.org/en/stable/classes/class_parallaxbackground.html