FAQ, pg. 9:

(3.19) Scope of Immunity
Immunity only protects a card itself. Peripheral entities attached to or associated with a card, such as attachments, duplicates, power counters , gold tokens on the immune card, and also including triggered effects originating from the immune card, can still be affected by cards of the type to which that card is immune, as long as the affecting cards do not target the immune card.

So, since Elaria does not target (ie, "choose") the character, immunity will not protect the power counters.