Shipping a client library and doing consumer driven contract testing sound like complete opposites to me

To some extent yes. So, do you think that an old school integration test approach will be more appropriate in this case?

The problem becomes when you have more than one client using the library, because that can lead to clients using different versions of the library. Do you choose to maintain integration tests to ensure coverage for every client version you ever released? (Maybe that is maintainable if breaking changes are minimal). Do you have a strategy for retiring old client versions now?