So my testing was flawed.
I onboarded my cell phone to the corporate WiFi, and it still didn't work, so I concluded that it couldn't be a firewall or border gateway issue. However, I didn't force-close the app or reboot my phone after changing networks.
Today, I did just that, and found that if I force close the app and relaunch while connected to the corporate WiFi I am able to use that drop-down. If I force close the app and launch while connected externally, it doesn't work. That tells me that the issue is entirely in our firewall or border gateway.
Any thoughts on what the issue could be more specifically? We're not seeing anything dropped in the firewall logs, and the MBG/MiCollab logs are so verbose that they're not really usable unless you already know what you're looking for.