Responding here with the solution I found. Hopefully it'll help anyone with similar issues.

First, the Datadog install script is practically a matryoshka doll- the script creates another script which creates a YAML file.
One of the consequences of that is that nothing from the original init script goes into the Databricks logs for init scripts. Instead, it writes its output to /tmp/datadog_start.log. The 2nd script that is created is also in the same directory at /tmp/start_datadog.sh, and I was able to use that to run it directly in a notebook and see the root cause of my error. The error itself was purely a syntax error, easily rectified when I was able to track it down.