The best approach in my opinion would be to have the Lightroom catalog on an external drive. This could be a thunderbolt SSD drive or an ordinary hard drive and then connect the drive to the machine you are using at the moment. The folders with your photos would need to be also switched between your two machines, of course. I believe that thunderbolt is not needed really and the real difference in using the catalog on different drives connected via thunderbolt or USB 3.0 would not be that much considering the amount of time doing different taks. I think you could try it out most likely without buying new stuff.
Using Dropbox I don't think would work well as the synch times would be very large given a reasonably large catalog and associated previews. I don't think Dropbox would be smart enough to synch just changed chunks of the catalog and previews as certain backup programs does. So therefore my recommendation to use an external drive.
Thanks. Would you also keep the Previews Catalog on the same external drive?
Bear in mind that I'm allergic to Lightroom lagging. Speed is absolutely essential.
Would the catalog (and preview catalog) on an external SSD USB3 result in even the slightest lagging compared to keeping them on the internal drive?