If you put the extra files in the data subfolder of your sketch they will be copied along with the source files when you save as.
Although I don't take this as evidence that you've outgrown the Arduino IDE, I do strongly recommend that you start to use Git. It's not something to inflict on a beginner in their first month of programming but clearly you're far beyond that stage.
I don't see any benefit to an IDE with Git integration. Instead, I would find a dedicated Git client program that you like (unless you're happy with using Git from the command line exclusively, then don't bother). That will be much more flexible than hoping to find an IDE that happens to provide the development features you want as well as the version control features you want. I started with GitHub Desktop (which works for local repositories as well as repositories hosted on GitHub). At that time, it was a very nice beginner-friendly program, but limited in features. I know they've done a lot of development on it since I quit using it so I can't vouch for the program in its current form, but I suspect it's still good. I'm now using Git Extensions, which is excellent but unfortunately currently Windows-only.