Problem Reinstalling Mac on Temporary External SSD - 'You may not install to this volume because the computer is missing a firmware partition'

Unfortunately the macOS installer requires a properly working and erased internal drive for the system firmware updater. It is a very frustrating requirement even in the best of times. Click on the "Contact Support" link in the upper right corner of this page to see if you have the option to mail the laptop to Apple for the repair. The mail-in repair option is available in some regions.

Have you tried running Disk Utility First Aid on the internal SSD to see if the file system needs repaired? Have you tried reinstalling macOS on the internal SSD? Or performed a clean reinstall of macOS on the internal SSD by first erasing the physical drive as GUID partition and APFS (top option)? To see the physical drive you may need to first click on "View" within Disk Utility and select "Show All Devices" before the physical drive appears in the left pane of Disk Utility.

Another option would be to boot into Internet Recovery Mode using Command + Option + Shift + R to boot the installer for the OS which originally shipped with your laptop which I believe may be macOS 10.12 Sierra which may not have or even need to update the system firmware so it may be possible to install to the external drive if the internal SSD cannot be properly erased or repaired. While this option is not perfect at least it will allow you to use the computer from the external drive until the laptop can be repaired.