Well I know it wasn't related to the iPhone 7 because I didn't even have XIM Manager installed on that particular phone yet. I had to install it so I could make a WWII profile. My previous device has been turned off for weeks so I know that's not it either. I'll try reinstalling the app and see if I have any better luck with it.
EDIT: No joy. I put the iPhone 7 into Airplane Mode and made sure the app was completely closed. Then I reinstalled the app on the X, toggled Bluetooth and Airplane Mode, and the XIM4 still just pulses blue to green every second or so while holding the Connect button. This is not the red/blue/green pulse that indicates corrupt profiles -- red never comes up. And as I said, the profiles work perfectly fine when paired with the iPhone 7.
I'm not sure what else to try, but for what it's worth, my X is the A1901 GSM model (D221AP board). Maybe it uses different Bluetooth hardware than the one tested, as I know the A1901 uses an Intel chipset while the A1865 uses Qualcomm. It's probably not related to that, but figured I would note it.