*This is more of a comment to Misha's question, but it does not fit as a real comment.*

The Lemma 4.2.2. in http://www.math.wustl.edu/~sk/books/root.pdf is not true for general complete doubling metric spaces nor is the Besicovitch covering theorem:

Take for instance a space $X = \mathbb{N} \cup\{0\}$ with the distance
$$d(0,j) = 2^{-j} \text{ for } j \ne 0$$ and $$d(i,j) = 2^{-j}+2^{-i} \text{ for }0 \ne i\ne j \ne 0.$$

To see that this does not satisfy the Lemma nor the Besicovitch covering theorem consider for any $k \in \mathbb{N}$ the collection $\{B(j,2^{-j}+2^{-k}) ~:~ j = 1, \dots, k-1 \}$.

**Note that this is not a counterexample to the original question as it is not a length-space.**