1) There may or may not be a final cause. (You don't need the anthropic principle to say this)
2) The possibility of a final cause is completely unrelated to the ability to ask if there is a final cause. (So no, the anthropic principle is completely irrelevant)
True, the anthropic principle does not say "there is a final cause" or "there isn't a final cause". However, as I argued, it does say that "it is not the case that there must be a final cause". In other words, "a final cause may or may not exist, but its existence is not a necessity". Do you agree with this at least?
Your argument critiqued
1) You irrationally assumed a Final cause from the existence of the survival instinct.
No support given
2) You propose the theory the assumed Final cause was a result of a designer.
Unreasonable start. Almost a strawman.
3) You use the anthropic principle to explain why the survival instinct is not evidence in support of an Efficient cause.
Valid but axiom 3 already covered it.
4) You conclude that since a proposed theory is not proved, then the lack of a Final cause is possible.
Merely evoking the anthropic principle was insufficient to eliminate the Efficient cause theory you proposed. (Axiom 3 is required since aliens pass Axioms 1&2) Furthermore eliminating the Efficient cause theory you proposed would not be evidence that there might not be any Final causes.
A better argument is:
There is no evidence that Final causes exist by necessity. Therefore we cannot conclude that a Final cause must exist. As a result of the binary nature of existence, if we cannot conclude a Final cause must exist then we adopt the position that it is possible a Final cause might not exist.
Notice how this avoids using any of your 3 axioms.
