First, I'd stabilize the footage 100%, and then from there you have a couple of options...
1. Get some B-roll handheld footage - of the proper motion you desire - of a white piece of paper with a black dot on it. Track the dot and apply its motion to your stabilized footage.
2. Utilize the wiggle expression on your clip's position (or a null object that you can parent the footage to).
I'm sure there are a few other ways to approach this, but that's where I might start.