Steam – Using Steam backup to store game files and saves on different drives

According to this article on the Steam support site:

Important:

Backups for Valve games created with this feature will not include saved games, custom multiplayer maps or configuration files.

So to answer your question, it will not back up your mods. However, the article goes on to describe how you can back up your full installation.

I think the following is the correct way to currently do what I wanted:

1. Ensure that Steam thinks the game is not installed. Close Steam just to be sure.
2. Copy the game files to SteamApps\common\<game_name>, where <game_name> is to be found out via googling/equivalent. In my specific instance, this was Batman Arkham City GOTY.
3. Launch Steam and install the game. The installation will go through a "Discovering existing files for _" stage.
4. Watch and wait as it downloads only the missing parts.

I still have to download 9.8 GB though (instead of 17) in my specific case.