Not sure why you think a BEC should fix it when a capacitor really didnt do anything.
In any case, its a process of elimination IMO. Since the electronics are new, I would start with the receiver and see if that changes anything. Even a different Tx if you can come up with one.
Then maybe swap a motor, a ESC and a servo to see if you can find the culprit.
Does the brownout happen right away when you start driving or after a few minutes? Can it be done on a stand?