I don't know the architecture of the system to say for certain yes or no, but depending on where the setting file is stored it's possible swapping out the APIM wouldn't reset anything
A quick search and I found the following:
The SYNC computer, which Ford calls the Accessory Protocol Interface Module (APIM), is housed separately from the
head unit, called the Audio Control Module (ACM), and interfaces with all vehicle audio sources as well as the high-speed and medium-speed vehicle
CAN-buses.
Source
If this is accurate (sounds like it is) then replacing the APIM wouldn't reset any settings, that would all be in the ACM. Now the question is could it be something in the ACM that causes this problem?