To follow up, I reported the issue to Chris Young (Mr. Stratux), in case he wanted to track production issues with the boards. He pointed me to the Stratux WebUI "Settings" page, and the "Calibrate AHRS Sensors" button on that page, which apparently does something different than the "Reset/Level" button on the GPS/AHRS page.
After clicking that button (with the Stratux on a level surface), the board I thought was faulty started behaving correctly.