Back up other drive using Time Machine
Time Machine is optimized to create and maintain a history of file-level backups of all files/folders on your system. What you rather need is a disk/image level backup which just dumps the whole drive onto an external drive. You can afterwards either copy the whole image back again or, probably more useful, get specific folders back.

The two applications people tend to use for this are CCC and SuperDuper!. You also need an additional external USB drive at least as big as your internal drive.