Okay, so having done minor testing this seems to work. I renamed the Immortal trait entries from this mod from "immortal" to "scmimmortal" in the following files:

common > traits > cheat_traits.txt - Find and replace "immortal = {" 1 entry, straightforward.
decisions > cheats_menu.txt - Ctrl+F and search for any "trait = immortal" Should be 6 entries in that file.
decisions > cheats_menu_intrigue.txt - Ctrl+F find and replace "remove_trait = immortal" 1 entry.
interface > cheat_traits.gfx - Find and rename "GFX_trait_immortal" 1 entry.
localisation > traits.csv - rename "immortal" and "immortal_desc" Only those 2 entries.

Those changes should allow both immortal traits (the one from this mod, and the one in the base game) to exist in the same game without any weird problems.