Numerical calculation in Mathematica gives $$ \sum_{k=1}^{\infty}\sum_{j=0}^{k}\binom{k}{j}\frac{B_{j+k+1}}{j+k+1} =0.069591059035995961110566767049... $$ This quantity is positive, while $\frac{2\,\log\phi}{1-2\phi}=-0.4304089410...$ is negative. So the identity under consideration is wrong.
Using the integral representation of Bernoulli numbers I obtain formally the integral representation of the double summation $$ 2\text{Re}\int_0^\infty\frac{t}{e^{2 \pi t}-1}\frac{1+i t}{t^2-i t+1}dt=0.069591059035995961110566767049... $$