After much groaning and gnashing of teeth I have come up with a successful solution.
rds_data (as confirmed by AWS Support) is only supporting Aurora Databases. Wish the AWS documents mentioned this. 8{(>
Thanks to the answers above as well as Jason Landrey for hints as to the solution.
In order to access RDS/Oracle, you need to use cx_Oracle. But wait, there's more.
cx_Oracle is not in the standard Lambda environment, so you need to bring your own. My development environment is on Windows, but the Lambda environment is Linux. So, you need to download and install in your packaging directory I got mine from https://pypi.org/project/cx-Oracle/#files. Install locally with:
pip install cx_Oracle-7.1.2-cp37-cp37m-manylinux1_x86_64.whl -t .
You will see several file appear in . Then you need to find a Linux system and download /lib64/libaio.so.1.0.1 and call it libaio.so.1 in your packaging directory.
And then you need to download both Oracle instant client basic and SDK packages from http://www.oracle.com/technetwork/topics/linuxx86-64soft-092277.html.
Create a zip file with all these items (including your own Python source). In doing so, rename Oracle instant client files libclntsh.so.11.1 to libclntsh.so and libocci.so.11.1 to libocci.so.
Upload the zip to a S3 bucket as the direct deploy is limited to 66mb and this zip is a bit larger.
Create a Lambda with the appropriate IAM permissions and VPC access, install the package and it should be good to go.
I found that if you don't include all the instant client files you start getting Oracle errors about missing timezone and NLS information.
List of zip contents (for me, YMMV):
7996693 08/24/2013 12:30 libnnz11.so
0 03/11/2019 16:10 cx_Oracle-7.1.1.data/
0 03/11/2019 16:10 cx_Oracle-7.1.1.data/data/
0 03/11/2019 16:10 cx_Oracle-7.1.1.data/data/cx_Oracle-doc/
0 03/11/2019 16:10 cx_Oracle-7.1.1.dist-info/
1325 03/13/2019 12:35 Email.py
1805 02/19/2019 21:11 cx_Oracle-7.1.1.data/data/cx_Oracle-doc/LICENSE.txt
163 02/19/2019 21:11 cx_Oracle-7.1.1.data/data/cx_Oracle-doc/README.txt
851 02/19/2019 21:11 cx_Oracle-7.1.1.dist-info/METADATA
628 02/19/2019 21:12 cx_Oracle-7.1.1.dist-info/RECORD
109 02/19/2019 21:12 cx_Oracle-7.1.1.dist-info/WHEEL
10 02/19/2019 21:11 cx_Oracle-7.1.1.dist-info/top_level.txt
2270301 02/19/2019 21:11 cx_Oracle.cpython-37m-x86_64-linux-gnu.so
2140 03/13/2019 14:21 getSecrets.py
5560 03/12/2019 08:48 libaio.so.1
53865194 08/24/2013 12:30 libclntsh.so
118738042 08/24/2013 12:30 libociei.so
7633 03/13/2019 16:39 scheduleReports.py