Just thought I'd update this.
So I finally brought in my 6S for the free battery replacement two days ago. I showed up, told the Genius my phone was eligible for the battery replacement program, and also mentioned the poor performance I've been experiencing. The Genius ran diagnostics but nothing appeared out of the ordinary. She said I could pay for a replacement phone but since I'll be upgrading in a few months I figured I could deal with it and I'd just have the battery replaced for now.
Couple of hours later and my 6S has a new battery in it. I've used it quite a bit since then and it has been running buttery smooth. I just ran another Geekbench test, here are my scores:
Metal: 9978
Single-Core: 2527
Multi-Core: 4385
My last benchmark, ran on July 22nd, gave me a 1456 single-core score and a 2469 multi-core score.
I have no idea why replacing the battery would make the phone feel like new again, but sure enough that's what happened. Just a heads up in case anyone out there with a 6S is experiencing crap performance like I did.