It has come to my attention that this example has been computed independently (and more directly) by Safia Chettih and Daniel Lütgehetmann. The result is given as Proposition 4.3 in v2 of their preprint

https://arxiv.org/abs/1612.08290v2

which appeared in July 2017. They use a combinatorial model due to Swiatkowski to show that $C_3(B_3)$ is a surface of genus 13 glued from 144 2-cells.