I am exploring the idea of moving/cloning our repos from github to VSTS. Currently code is checked in through to Github. The advantage I would get is that all the teams would be using VSTS and all the bells and whistles offered by VSTS.
The second option is definitively the best one for you:
VSTS Git repositories are just plain standard Git repository you could reach by means of any Git client using authentication over SSH and/or HTTPS to push/pull from it;
VSTS even allows you to use SSH key authentication
there are Credential Managers available for all the major platforms you could use to authenticate easily;
no useless hassle syncing two repositories, why would you do that? :-)
I certainly suggest you to try the Oxygen Git plugin to connect to a VSTS Git repository and verify that it works as you would expect.