BF1 is kind of a hard call. I know Techspot had trouble testing as many configurations as they would have liked, because it kept locking the accounts after so many "hardware configuration changes" -- IIRC, they even had trouble when they tried to artificially downclock their Skylake i7 for testing purposes. Basically, they weren't even able to test with Haswell chips (let alone Skylake).
Having said that...in most other games released in the past 12-24 months you see more differences between Intel chips of the same class due to their core clock speeds rather than the generation they belong to. In most of those games, an i5-2500K (about 99% equivalent to the i5-2500 you're asking about) holds its own with Haswell & Skylake core i5s (&, for the most part, core i7s), losing maybe 5% tops on performance with the same GPU, & the times where it lags behind the core i7 chips are the times that the Skylake i5 chips also lag behind the core i7s. In fact, in a lot of those games its performance is equal to or slightly better than the higher-clocked i3-6100 (3.7GHz version).
So, yeah, it should be able to run 60FPS on medium/high at 1080. In fact, based on how the GTX 1060 performed with a Skylake i7 (http://www.techspot.com/review/1267-battlefield-1-benchmarks/page2.html) & how the i3-6100T (3.2GHz) performed (http://www.techspot.com/review/1267-battlefield-1-benchmarks/page4.html), you should be able to hit at least 70FPS on Ultra with DX11, possibly up to 80 with DX12 (although apparently it won't run as smooth).
The question is whether you want to buy parts for an older system. If you're not worried about its future upgradeability, don't need more advanced features (i.e. built-in M.2 ports, DDR4 RAM, etc.), & the price is cheaper than a Skylake build, then go for the i5-2500. Otherwise, you might want to consider buying newer parts.