It's a Virgin phone, but I figured out the code, it's SPRINT (777468). Unfortunately, there's just these options:
1X Engineering
Configuration
Bluetooth
Data Status
Memory

All just display info and don't accept any input or toggle any settings.
I think ##3424# that displays a menu with options DM and OFF is probably the DM mode I'm looking for. When in that menu, DM is always selected by default and when I press the key corresponding to OK or even if I press BACK, the phone reboots. At first, I wondered if the reboot means it's rebooting into DM mode, but since DM is selected by default even if I previously chose the OFF option, I'm starting to think the reboot means DM mode is disabled in firmware.