I think the problem with your BF1 performance is more on the GPU side than with the CPU. Not that there might not be a CPU issue, but CPU-wise BF1 makes even Skylake i5 & i7 chips struggle.
http://www.techspot.com/review/1267-battlefield-1-benchmarks/page2.html: Even with an i7-6700K, they only managed to get to 45FPS with an R9 270X @ 1080p resolutions. Based on how the CPUs tested during the beta (https://forums.anandtech.com/threads/battlefield-1-beta-cpu-scaling-performance.2485172/), I could see that dropping down to maybe 30FPS.
In other words, you'd see maybe a 30-33% increase maximum by going to a Skylake build. In contrast, replacing your R9 270X with an RX 480 or GTX 1060 (6GB model) would apparently double your performance...more than enough to be able to hit 60FPS @ 1080p resolutions. That would run you maybe $250 USD tops, a much cheaper upgrade than replacing your system & starting over from scratch.
If you're still worried about the CPU, though, let us know what motherboard it is. Right now, you still have a very good CPU -- 2nd-tier on the list (http://www.tomshardware.com/reviews/cpu-hierarchy,4312.html), right behind the Ivy Bridge/Haswell/Skylake core i5 & i7 chips (& tied with AMD's 8-core FX line). That's not bad for a 5-year-old design. But if you're worried about your CPU horsepower, & your motherboard can handle it, I would strongly recommend getting one of the Sandy Bridge (or Ivy Bridge, if it will work) i7 chips.