So Ive read this

The regular exit is reached by the standard method of pressing up on the d-pad when in front of the door. The alternative exit is reached by instead entering the door by pressing down. Most of the alternative exits work as shortcuts, but not all.

However pressing down does nothing?! Am I doing something wrong? Is the door time based as I need to get to it by a certain time?

If playing the English version, hold B and enter the door pressing up.

On the JP version, press down!