time machine can't backup internal HD when booted from external drive

Hi

Hardware setup

1. iMac (Yosemite) (SMART shows failing) - no longer boots, gets stuck half way into apple logo. Can access single user mode but see i/o errors with fsck -fy and other commands show the same trouble.
2. 1 external USB to boot OS X (Capitan - boot USB created using a MacBook Air)
3. 1 external USB drive for TimeMachine (wiped clean only a few days ago - backup no longer exists - was planning a fresh new backup **bad timing**)

Note: When booted from external USB, the internal "Macintosh HD" drive mounts and I have already backed up the most critical data.

What I would like to do but am unable to - backup internal Macintosh HD using the Time Machine feature to the same Time Machine drive (which was wiped a few days ago), but it does not show up in the Time Machine System Preferences, and as a curiosity I can't exclude it either. Is this expected behavior?

If this is expected I will use Disk Utility to create an image or even a bootable drive.

Thanks for any info.