yes there have been known issues with external drives. The external drive must be connected & mounted before opening LW. Sometimes LW has problems reading or seeing external drives when it opens & the download or shared files folders location reverts to default location.

Your external drive needs to be fast. It helps if your cpu is also fast. Too many incomplete files to load when opening can also cause LW problems finding an external drive.

It's worth a try though. So .. how to to:
Within LW do as image above in post #4 suggests but instead direct LW to your external drive. Then close down LW. Replace the new download & incomplete folders with the new ones. But make sure the download folder has the same name as the other. This should work. Just in case of problems, make a backup of your Downloads.dat & Downloads.bak files found in the incomplete folder before you begin this process.

Personally I have multiple internal hard disk drives & choose one of those instead for my downloads & shared files. I don't like storing user files on my OSX system drive user account so I can maintain as much free space as possible. The more free space the system drive has, the better it functions. All OSX programs & also the system uses VM (Virtual Memory) storing temporary memory on the system drive regardless of how much ram you have.

And on the topic of VM, OSX's Java uses a lot of VM & unfortunately has for a long time had bugs releasing this VM when no longer needed. Thus the longer you use LW, & the more searches you do, the more incompletes you have, etc. the more VM is taken up by LW over a period of time until you close the program.