I run an internal/external setup and it works, but I had to remove the internal SSD before installing on the external SSD.
What you are doing, in a nutshell, is creating a portable Solus installation on the external drive. I boot into th external drive on various computers (including my Solus computers) as a portable Solus installation as needed. All I have to do it plug it in and boot the drive via the BIOS boot menu (F12 on my Dell computers).
If you can remove the internal SSD, these are the steps:
(1) Remove the internal SSD (the one on which Solus is currently installed with UEFI) from the computer.
(2) Plug in the external SSD.
(3) Boot from your installation Live USB.
(4) Because the external SSD is the only drive available at this point, the Live USB should recognize the external SSD as a target drive.
(5) Install Solus on the external SSD using a USB as your installer.
(6) Put the internal SSD back into the computer. When the external drive is plugged into a USB port, the computer BIOS will recognize the external drive as a bootable drive, just as it does a bootable USB drive.
At this point, you will have two Solus instances installed, one on the internal drive and one on the external (portable) drive, each drive having a distinct UEFI boot partition and UEFI boot loader.
You will need to use the BIOS boot menu (F12 on Dell computers, maybe something different on yours) to access the drives on startup when the external (portable) drive is plugged in. Linux Boot Manager, which controls the internal drive, should be the first boot loader in the BIOS boot menu, and the boot loader controlling the external drive should be the second boot loader in the BIOS menu. The boot loader for the internal (primary) drive is identified as "Linux Boot Manager" and the boot loader for the external drive (portable) is identified by drive identification -- e.g. PNY1311-171722098 in my case.