you're going to need something like msi afterburner to check the usage of the cpu. it also sounds to me like you're bottlenecking at the cpu. play the game for a bit, be sure to encounter these low fps you're talking about, and see how it fared. if you're topping out the usage of the cpu, then its likely bottlenecking you. also watch the temps, its possible it could be throttling, but thats not really the behavior you seem to be experiencing.
i would also try cutting down settings that have little visual impact but save you a lot of performance. things like anti-aliasing, shadow quality, and complex lighting. im not sure where the game benchmarks on a single 970, so i dont know what settings you're supposed to be aiming for or what resolution you're trying to run at.
you should also look up similar systems running the game, often googling "a10-7700k gtx 970 battlefield 1" will net you some youtube vids of people benchmarking with that or similar configurations, so you know what you should be looking for in performance.
lastly, i'd be sure your system isnt trying to share the resources of the apu and gpu as graphics processing. unsure how well it will work, but i do know its rare for unusual configurations like htat to be specifically worked on by the devs to be sure they work the way you'd want. try running it with just the 970 working at the graphics.