To use the HDD as a backup drive, you will have to erase the disk. Time Machine needs the whole hard drive or partition to make backups.
Alongside with Time Machine backups, I recommend you to make a second partition on the HDD and make clones with Carbon Copy Cloner or SuperDuper. This will simplify the restore process if your SSD gets damaged or you have a problem, so you will only have to connect the HDD and boot from it to use the Mac
First get an enclosure for your HDD. How did you clone your HDD/SSD? If you used Carbon Copy Cloner then you should set it to update those changes that you make to your internal SSD. It will be a bootable HDD and can be swapped any time.
if you decide to Use Time Machine, it will not be a bootable drive but it will be the most efficient backup device. The best of all possible worlds is to have both.
I can assure you that Time Machine does not create a cloned backup. For that you need other backup software like Carbon Copy Cloner. Disk Utility can also create a clone, but DU cannot be used as ordinary backup software. If you want a backup that can be used as an emergency boot drive, then don't use Time Machine for your backup.