You are correct, they aren't logged on your DC's. However, I believe (but am not positive) that if your auditing is high enough on the DC's, an event is logged when someone disconnects a network drive. So if you are mapping drives for your users, you could potentially track their log offs but looking for that 'dismount' event. Even then, I'm not sure I would want to rely on that. You said you can't deploy scripts, but I'm just mentioning that in case you already are in your environment.

For Git Bash

If you are running msysgit (I am assuming you are) and are looking to run Git Bash (I recommend it over TortoiseGit, but I lean to the CLI more than GUI now), you need to figure out what your home directory is for Git Bash by starting it then type pwd (On Windows 7, it will be something like C:\Users\phsr I think). While you're in Git Bash, you should mkdir .ssh.

After you have the home directory, and a .ssh folder under that, you want to open PuTTYgen and open the key (.ppk file) you have previously created. Once your key is open, you want to select Conversions -> Export OpenSSH key and save it to HOME\.ssh\id_rsa. After you have the key at that location, Git Bash will recognize the key and use it.

Note: Comments indicate that this doesn't work in all cases. You may need to copy the OpenSSH key to Program Files\Git\.ssh\id_rsa (or Program Files (x86)\Git\.ssh\id_rsa).

For TortoiseGit

When using TortoiseGit, you need to set the SSH key via pacey's directions. You need to do that for every repository you are using TortoiseGit with.