I'm being 100% honest when I say that the i5 is the best option here in my opinion. It will run all his games perfectly. He clearly plays more than BF4. He said BF4 and other intensive games. It's not like this will be a dedicated BF4 build.
The i5 and 280x in my build will be perfect for his needs at 1080p.
I would DEFINITELY not recommend an i3 to someone wanting to play BF4 especially when pairing it with a GPU like a 280x.