As it turns out, the Z and C buttons don't register as the L and R triggers in Dream Passport (being leveraged for Bonus Mode). As such, there was no way to pull up the menu and exit back to the game. As a result, I spent some time hacking in a new universally compatible button combination that'll do the trick: A+B+X+Y

See a video of it in action here...
https://i.imgur.com/X26UL8V.mp4
I've also added this modification as an option to my publicly available hacked Dream Passport browser: viewtopic.php?p=151960#p151960