Oops, it seems I slightly goofed the part number, but I have the right datasheet linked. I should be TPS630701 instead of TPS63071. TPS630701 has Vout tied directly to the feedback pin, and the voltage divider is internal to the IC.
Per Wrapper's idea, a 100 Ohm resistor between Vout and GND did allow the display to start up, but a 470 Ohm did not. I measured 1.25 mA coming from the LEDs in sunlight. If I treat them as a constant current source, I would need at most 200 Ohms, preferably a bit less to ensure stability, to keep the voltage difference below 0.3V. Since Vout is a 5V when running, this will use about 0.125 to 0.25 Watts depending on the resistance chosen. I was hoping for a more power efficient solution since the display is running on batteries.