How to install 3rd party odbc driver at RDS sql server

I don't think I can install the ODBC driver because RDS is a managed service

In general RDS does not allow the user to customise the OS or DB environment ... but RDS Custom gives some of that power back if you need it - https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/rds-custom.html. It's best though to stick with standard fully-managed RDS if you can, it's likely to result in lower TCO.

Generally an ODBC driver would be installed in the DB's client though, so as part of your application?