After several hours of troubleshooting, I found this issue to be caused by having Oracle.DataAccess.dll (v4.0) in my projects bin directory, but the runtime also loading Oracle.DataAccess.dll (v2.x) from the GAC. Removing and readding the Oracle.DataAccess entry in the project references solved the problem for me.
The other files mentioned here did not appear to be necessary in my situation.
The root cause of the "The provider is not compatible with the version of Oracle client" error is (generally) that the managed assembly is attempting to load unmanaged libraries which do not match versions. It appears you can force the Oracle driver to use the correct libraries by specifying the library path in the web.config1
<configuration>
<oracle.dataaccess.client>
<settings>
<add name="DllPath" value="C:\oracle\bin"/>
<!-- ... -->
</settings>
</oracle.dataaccess.client>
</configuration>