Looks like it is a Jemmy feature/problem/bug after all. The JemmyProperties class has a static initialization block which gets to the previously linked code and prints the message anyway.
To get around it I decided to pipe it through some pattern searcher like findstr.
javaw -javaagent:"C:\<pathToProject>\lib\remoteswinglibrary-2.2.1.jar=127.0.0.1:1234:APPORT=1234" -cp "swexpl.jar;swag.jar;swinglibrary-2.3.0;RemoteSwingLibrary-2.3.2" -jar "C:\<pathToProject>\MyApplication.jar" 2> errors.log | findstr /b /v Using
I also noticed that the output was not generated when the application was tested by RobotFramework. I had to use the Set Jemmy Dispatch Model keyword with the QUEUE parameter.
So I have managed to complete the test.