So Gates Rudd, Dunfield, and Obeidin algorithmically obtained a surgery diagram in their preprint Computing a link diagram from its exterior. Here's Figure 21 from Section 9.2.
Actually I had made one back in 2019 based on the earlier answers of Moskovich and Ruberman and Agol's comment. For whatever reason, I had forgotten about it. Some simplification could be done, but then 5-fold reverberation gets lost.
Edit: So I went ahead and made a Sketches of Topology post giving some details about how this diagram was obtained.