The second and even greater mystery is that when I installed the Oracle client, it installed its own Perl executable. Should I be using this to run/install DBD?

No. It is usually old and not needed except to recreate parts of the Oracle software that are already installed on your computer. Try to get rid of it.

Oracle Installer is a pain, perhaps get rid of the entire Oracle installation and install "Instant Client" instead. As a rule of thumb, install only the ODBC driver (that depends on all required libraries and so also installs them) and everything that looks like "development", "libraries", "headers".

Alexander