There is a finitely additive, translation invariant measure on the entire power set of $\mathbf C$ (see, for example, [Amenability, the ping-pong lemma, and the Banach–Tarski paradox](https://terrytao.wordpress.com/2009/01/08/245b-notes-2-amenability-the-ping-pong-lemma-and-the-banach-tarski-paradox-optional)).  Clearly, it satisfies your desired condition (1).  Since rational-length, half-open ‘intervals’ in $\mathbf C$ are finite, disjoint unions of $1/n$-equal parts that are intervals for various $n$, and generate the Borel $\sigma$-algebra, and since Lebesgue (I would rather say Haar) measure assigns measure $1/n$ to a $1/n$-equal part that is an interval, we have that any countably additive extension as in (2) extends Haar measure.  I suspect that a Vitali-type argument shows that no extension as in (2) exists, but I do not currently see it.