Given that its a code-share through AA, and AA doesn't know about Evergreen Deluxe, AA is probably selling straight 3-class Y/C/F tickets and having them convert over to the equivalent BR classes. But, it doesn't hurt to check with BR directly to see if something can be done.

The EVA contact numbers can be found by clicking on the America link at EVAAIR Contact Info where you will see offices listed for Dallas and Houston on the pull down display. Also, you might want to check EVA's actual arrival times for your flight. You can check the recent arrival times of your flight (BR 11) by clicking on the Flight Info button.