After a bit of digging, I believe the issue was that the 9.6 install setup a cluster for the new instance which was interrupting the pg_upgrade. I ended up getting it working by dropping the new cluster and upgrading the 9.3 cluster using pg_upgradecluster from the following gist: https://gist.github.com/delameko/bd3aa2a54a15c50c723f0eef8f583a44
Install packages
sudo apt-get install postgresql-9.6 postgresql-server-dev-9.6 postgresql-contrib-9.6 -y
List clusters. Should show existing (9.3) and newly installed (9.6)
pg_lsclusters