Different than how yours is set up, I’ve long mounted attached volumes with a scheduled AppleScript since I would rather specify the times when Time Machine is run.
set theDisk to "Disk Name" do shell script "diskutil mountdiskutil list | awk ‘/ " & theDisk & " / {print $NF}’"
After Time Machine runs, another AppleScript unmounts the disk.
set theDisk to "Disk Name" do shell script "diskutil unmount " & quoted form of ("/Volumes/" & theDisk)