Asymptotics of shifted Catalan numbers

Question

I am trying to understand a lemma from a paper (most of the proof was omitted), and I've got it melted down to the following: Let $B_n$ denote the $n$-th Catalan Number. In a sufficiently small neighbourhood of $z = \frac{1}{4}$ we have \begin{equation} \sum_{p \ge p_0} B_p z^p = \frac{C}{\sqrt{\ln (\frac{1}{1-4z})}} + O\bigg(\frac{1}{\sqrt{\ln^3 (\frac{1}{1-4z})}}\bigg) \end{equation} where $p_0 = p_0(z) = \ln \frac{1}{|1-4z|}$ and $C = 2 \sqrt{\ln 4 / \pi}$. I guess one either has to express it as the difference between the generating function of the Catalan Numbers $B(z) = \frac{1 - \sqrt{1-4z}}{2z} $ and a finite sum of Catalan Numbers, or calculate a series of shifted Catalan Numbers. I've tried both; yet neither one of them yield the desired solution. The lemma has been taken from the paper Analytic Variations on the Common Subexpression Problem by P. Flajolet et al (Lemma 1). I asked a part of the problem in question 107773, where it was suggested that I ask the whole problem, not just a part of it. I would be grateful for any help.