Ramichaud:
You can unmount the internal drive by dragging its icon to the trash,
BUT...
...Now and then the Mac may choose to "re-mount" it again, on its own.
Here's something you might try (I have no idea if it will work):
Click on the desktop to bring the finder "to the front".
Go to the menu bar and choose finder/preferences.
In the box that opens, UNCHECK "hard disks" but leave "external disks" CHECKED.
Close finder preferences.
Does this make the internal drive disappear from the desktop?
And still leave the external boot SSD visible?
If so, that's what I'd go with.
If that DOESN'T work, or if it results in undesired behavior, my advice would be to "learn to live with" the internal drive icon on the desktop, even if you don't normally access it...