I don't think it is at all clear from the text of the Presidential Succession Act that an Acting President would be bumped out of office by a newly-appointed Vice President. Quite the contrary actually:
So if the Speaker becomes Acting President because of a deadlocked election, then they
would get bumped out once the Senate chooses a new Vice President. But if the Speaker becomes Acting President because the President and Vice President are
dead, then the act says they're supposed to finish out the term.
Now, as for the Constitutional authority for the Succession Act,
Article II, Section 1, Clause 6:
So again, it looks like the Acting President is intended to fill out the term. (This raises a question as to whether the provision in the Succession Act for a Speaker to bump a cabinet member out of the presidency is unconstitutional, but that's a whole other issue). I understand there's an argument that this is superseded by the 25th Amendment, but I really don't see how that could have been the intent of the 25th Amendment. The office of the presidency wouldn't be vacant, and it doesn't really make any sense that the Vice President would have a superior claim to the White House than the Acting President that appointed them in the first place.
