According to the very few things I have read from Ford on the BP subject, it's just that BP gave them some money to put the logo and "Ford recommends" on the gas cap. They did not design anything around it; it was simply a way to raise a few extra bucks. If Ford actually designed the cars to work only with BP, that would lead to a massive anti-trust lawsuit, and that would probably destroy the Mustang brand.
It would be self-defeating anyway because not all BP stations use the same pumps, and many non-BP stations use BP gas.
There was a recall (I believe) for this for a sort-of fix, so it should be covered by the dealership even past the warranty. Try making a video of the problem happening at a few different gas pumps (ones close to the dealer would be ideal), so they can watch the problem happen.
The difficulty with this is that the design problem is inside the gas tank, and that tank was designed never to be removed or modified. Anyway, find the recall or TSB, and let Ford deal with it.