I have some insight into this now. I did an strace of the process and the short answer is that there is conflicting posts online about the right way to do this.

When installing DBD::Oracle, the advice is to build it against the Oracle Instant Client so that database upgrades don't break DBD::Oracle. When you need DBD::Oracle *and* you need to connect '/ as sysdba', you can't use the instant client because you are requesting an internal connection to the database. This attempts to exec the $ENV{ORACLE_HOME}/bin/oracle binary which does not exist.

For this to work, you have to build/link against the full database install.