None of the WS-AT enabled web service calls would work and failed with the following exception...
This was strange because all of the code was dot-to-dot as per the documentation and the data source connection pool was set up with the oracle.jdbc.xa.client.OracleXADataSource driver. It should have worked.
I figured that there had to be some problem with the data source still, so I went looking around. And I didn't have to go far because the Transaction configuration tab revealed the problem.
The "Use XA Data Source Interface" option was not enabled! The fix was of course to enable this option and then restart WebLogic. Once that was done all WS-AT calls worked perfectly.
So the exception message may not have been very intuitive but the fix certainly was. It pays to inspect configuration on the managed server side when using new features that depend on the environment being configured correctly first. Lesson learned - don't assume that all configuration is going to be as expected!