I figured I would report back. We ended up using Bounce instead of Luggage Hero because they seemed to have a more extensive network and had higher insurance coverage. Using Bounce, we stored our luggage at a Chinese restaurant in Times Square, a custom leather shoe shop in Florence and a hotel in Rome. I was a little nervous at first, but the process was very smooth and everything went great.
Bounce only had daily rates whereas Luggage Hero offered hourly rates. However, Luggage Hero added fees Bounce didn't so the cost was roughly the same if storing for more than 3 hours.