Tough call, but I'm going to agree with the i5-4440.
BF4 Multiplayer is very CPU intensive so the i3 would likely throttle you to the point where the better graphics don't matter at times.
And yes, the 4-core rig will last a lot longer. Drop in a better graphics card in two years or so and it'll last a few more years.
As for the FX-6300 being comparable to the i5-4440 I completely disagree. It's not even close, the i5 is way better. There are lots of benchmarks that confirm this though just substitute the i5-3570K or i5-4670K since 95% of the games perform nearly identically with these i5's.
Such as:
http://www.bit-tech.net/hardware/2013/06/12/intel-core-i5-4670k-haswell-cpu-review/5
But where's the FX-6300? (it fell off the chart below the FX-8350)
If we're talking SPECIFIC games then the FX might do okay if it's more GPU bound, but definitely not on average. Also, BF4 benchmarks are often Single Player but Multiplayer is far more CPU dependent.