Thanks a lot for these (surprisingly) quick and helpful answers; they put me on the right way for my solution.
The codebase were I want to use this, uses java.util.logging as its logger mechanism, and I don't feel at home enough in those codes to completely change that to log4j or to logger interfaces/facades.  But based on these suggestions, I 'hacked-up' a j.u.l.handler extension and that works as a treat.
A short summary follows. Extend java.util.logging.Handler:
class LogHandler extends Handler
{
    Level lastLevel = Level.FINEST;
    public Level  checkLevel() {
        return lastLevel;
    }    
    public void publish(LogRecord record) {
        lastLevel = record.getLevel();
    }
    public void close(){}
    public void flush(){}
}
Obviously, you can store as much as you like/want/need from the LogRecord, or push them all into a stack until you get an overflow.
In the preparation for the junit-test, you create a java.util.logging.Logger and add such a new LogHandler to it:
@Test tester() {
    Logger logger = Logger.getLogger("my junit-test logger");
    LogHandler handler = new LogHandler();
    handler.setLevel(Level.ALL);
    logger.setUseParentHandlers(false);
    logger.addHandler(handler);
    logger.setLevel(Level.ALL);
The call to setUseParentHandlers() is to silence the normal handlers, so that (for this junit-test run) no unnecessary logging happens.  Do whatever your code-under-test needs to use this logger, run the test and assertEquality:
    libraryUnderTest.setLogger(logger);
    methodUnderTest(true);  // see original question.
    assertEquals("Log level as expected?", Level.INFO, handler.checkLevel() );
}
(Of course, you would move large part of this work into a @Before method and make assorted other improvements, but that would clutter this presentation.)