Are you aware of limitations around site support with this approach?
Several years ago I implemented a similar feature just reversed - A remote machine logs a user in, then passes the cookies that result from login to an extension running in the user's browser, which drops them into the browser's cookie jar.
Worked very nicely, right until you run it to log into a service like GMail.
Then Google correctly notes that you're using the same cookie in two different locations, assumes you've been session-jacked (and you have, really - you just did it willingly), and locks EVERYTHING. It took a notarized copy of my drivers license before they let me back in.