So Gates Rudd, Dunfield, and Obeidin algorithmically obtained a surgery diagram in their preprint [Computing a link diagram from its exterior][1].  Here's Figure 21 from Section 9.2.[![enter image description here][2]][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. 
[![enter image description here][3]][3]

Edit: So I went ahead and made a [Sketches of Topology post][4] giving some details about how this diagram was obtained.


  [1]: https://arxiv.org/abs/2112.03251
  [2]: https://i.sstatic.net/2ZWo1.jpg
  [3]: https://i.sstatic.net/NbSKa.jpg
  [4]: https://sketchesoftopology.wordpress.com/2021/12/09/a-sketchy-surgery-description-of-swd/