Can I exclude files or folders from external drive in Time Machine

Time Machine may be confused which drive is which. Perhaps turn off Time Machine, determine which one has the backup dated folders created by Time Machine, and rename that "My Time Machine" drive, so when you go to select the additional files from the other drive, you are sure to select the right drive.

Is the external device Sunll you have hooked up the drive that the Time Machine data is backed up to? That may be why it is grayed out. You can only backup to a different drive for obvious reasons.

Go to Apple menu -> System Preferences -> Time Machine -> Options to add folders for exclusion.

Turn off Time Machine before you do it.