By default, /var/atlassian (and all subdirectories and files) would NOT be included in your backup. Out of interest, the default inclusions/exclusions are noted in the Core profile. All appliances have their own profile (which inherits from core), but as the Core config is pretty good, most of them are empty, e.g. LAMP. A few do have explicit inclusions and/or exclusions though.

So if you want to add (or exclude) files and/or directories, you can do that via the overrides file (/etc/tklbam/overrides). The default example file should include plenty of info about how it all works, or you can also view the example overrides file on GitHub.

Essentially just note the absolute path (on a new line) to include a directory (or file) and then it will be included in the backup. If it's a directory, then all the contained sub-directories and files will also be recursively included. To exclude a file or directory, it's the same, but proceeded with a '-' (i.e. dash or minus sign). Inclusions and exclusions can be mixed and matched to pin point specific paths to keep or not. In the case of conflicts, later overrides will override earlier ones. To provide a bit of an example:

/path/to/include
-/path/to/include/but/not/this
/path/to/include/but/not/this/except_this

Would include '/path/to/include' (and all it's sub-directories and files), but exclude '/path/to/include/but/not/this', except for '/path/to/include/but/not/this/except_this'. I hope that makes sense...

If you have a new enough version of TKLBAM (I foget when it was added) you can test your overrides by running a simulation, like this:

tklbam-backup --simulate

Then you can poke around in /TKLBAM to see what's in there...

Please note too, that whether or not you'll need to reinstall 3rd party software will depend on how that software was installed. If it was installed via a third party apt repo, then it will need to be manually re-installed (the repo will still be configured, just the package won't be auto pre-installed). SImilarly, if it was installed via some other package management system (unless of course you explicitly include the files in your backup). If you want, you can create a TKLBAM hook to do that for you, but I won't go into that here...