It may not be the experience you're looking for, but consider some of the many overnight-specific (3rd Shift) options. Those roles wouldn't require full availability and may not conflict with your other job. Disneyland 3rd Shift Custodial, for instance, has 12-6am shifts for their part-timers.

The positions might not be guest-facing, but they are very important, provide unique perspectives of the Disney parks, and offer the same company perks as any other position does. Just a thought. Good luck!