I would start simple....

More often then not the BEC brownouts can be smoothed away with just a glitch buster. I am not sure if what all you have is hooked up properly not having seen it. But if you are willing to gamble $5-6 at it just get a glitch buster and remove the external BEC from the setup. This will mean you need to make sure all three wires on the ESC are plugged into the receiver though (should have that middle red wire disconnected when using an external BEC).

Or if you plan to run a higher receiver voltage than the MMP can do, or simply have zero faith in a glitch buster let's make sure things are all hooked up right. You simply should not have this issue with the external BEC. First off the middle red wire to the ESC needs to be disconnected, that is where it would want to feed power to the receiver and take on the role of BEC. Beyond that it is simply a matter of properly attaching the external BEC you have to things based on their instructions.

I am not a personal fan of the external BEC though, I like simple and tidy and they just end up being more wires and parts in the chassis to deal with. The glitch buster is not a whole lot better in that respect, but it is simpler and there is only about half as much of it to deal with physically in the chassis where space is valuable if I had to do it at all.

Also, of the three players in this brownout game (Battery / BEC / Servo), the battery really plays the smallest part.