Honestly, I think putting any files (but especially the catalog file) on a flash drive for usage like this is a dangerous way to go. You'd be better off getting your synology up and running and putting everything but the catalog and previews on the synology. The catalog file ought to remain on one of the 2 external drives, and said external drive should be moved from computer to computer to allow you to access the exact same catalog file from both computers.
Spider Oak and Dropbox and similar services can be used for all files, although you may suffer a speed penalty for having your files in the cloud, and you'd have to be sure everything has updated before you use the files.
If/when you move your photos, you must keep the directory structure the same as it is now. You do NOT move photos or folders around within the existing directory structure. If necessary, you reconnect your Lightroom catalog to the new location of the photos, see Adobe Lightroom - Find moved or missing files and folders