Sync issue - Not Backed Up, Corrupt or Missing Original

You mentioned using an external hard drive with the two computers. Any chance you are trying to use the same drive between the two computers? That can be problematic. 

If you haven't tried this already, I'd suggest trying to sync everything down on Windows (your 2nd computer) without using the local storage option, just to make sure you can get the same number of files synced on both computers. Then you can enable the preference to store backups locally. But it will need its own local backup, independent from the Mac.

The details of what you've done and what is going on are a little bare here. It would be great if you could provide a detailed description of what was done in this "set up a new instance" operation. 

Generally, when this error occurs, it is advisable to go to lightroom.adobe.com and look for a Sync Issues folder. 

 The thumbnail will usually identify the offending surface and allow you to attempt recovery from that device.