Rayami Mutated has a different name
From the article


  Some creatures, like Bristling Boar above, have abilities that refer to themselves by name. Remember that from a rules perspective, a card using its own name in an ability really just means "me." For example, if Cloudpiercer ends up on top of Bristling Boar, that ability will work on the resulting creature.


As per Rule 201.4:


  Text that refers to the object it’s on by name means just that particular object and not any other objects with that name, regardless of any name changes caused by game effects.


From this it is clear that the second ability will apply to the resulting creature, even if it is not named "Rayami, First of the Fallen"