It could be a permissions issue. Highlight the folder on the Seagate drive that you are copying into. Press Cmd+i to bring up the info dialog and ensure you have Read & Write permissions. If not click the padlock icon to change.
It is the Location you have chosen, IE the folder on the drive. Create a new folder on that drive called Lightroom Backups and point LR to that folder to do its backups to. In Finder Right click on that new folder you created and make sure your User Name has both Read + Write permissions.
Also it is a bad idea to have one drive for Time Machine and to store other files on that drive. Time Machine should have it own dedicated drive to do its back ups to that is not used for anything else.
Just Shoot Me
On using Time Machine.
TIme Machine lets me make version end copies automatically to different hard drives .
i use it that way to back up everything on my Mac Pros.
That way I have redundant copies should a hard drive fail.
I also use Carbon Clone to make a full clone of my Mac 1TB Internal hard drive including a recovery disk every week and store it on a separate Lacie external hard drive.
I was just repeating what I read when I used a Mac. It is best to have a dedicated, or more than one dedicated, drive for Time Machine. This or these drives are only used for TM backups and don't hold any other files.
In older versions of OS X, up to Mavericks as that was the last version of OS X I used, TM likes too take control of the drive to make sure it has full access to all the space available. Maybe that has changed with newer versions of OS X. A great resource for info on Time Machine is the Pondini.org website.