The only possible option i can think about is to keep your Catalog and files on an external drive, or keep your files on a network drive and catalog on an external drive or Computer hard drive.
Check this link for more details.
- Sync LR Catalog between 2 computers but not master files
- The Easiest Way to Share Lightroom Catalogs with Different Computers - Lightroom Killer Tips
Copying the catalog and previews to the other computer's hard drive using the USB stick should work. Having LR open the catalog on the USB stick may be problematic.
The easy solution you are looking for really is the external drive to be swapped between the computers as needed.
I do a simillar thing! But the other way around: I copy all on an usb3 external disk. I sync the catalog to my laptops internal SSD and also to my PCs internal drive. I then can take my laptop with me and work on smart previews and leave the source files at home. When I am back I plug the external disk into the laptop and all is updated. Then I sync the catalog and am good again.
Syncing comes at a price. I used Synctoy from Microsoft which is not very transparent and seems to confuse echo and sync. I lately changed to Allsync which does a good job. There is always the potential of a disaster if the clocks are not synchronised. Or the elder files are somehow touched and then appear younger. A straight copy of the catalog seems almost faster than syncing. Usb3 speed for an external HDD is good enough, usb2 for flash drive is too slow. My Usb3 flash drives seem only to do usb2 speed.