Not sure why you're comparing to a smaller phone. We need more benchmarks, but it's not likely to achieve better battery than the S7 Edge (US version). It definitely will not beat the S7 Edge (Int'version) as this is holy grail of Android battery life right now. Both versions of the Galaxy S7 not only have a great camera, but it's considered to be one of the world's best phone camaras (but you've already acknowledged that it's better).
The S7 Edge is waterproof, has wireless fast charging, and Samsung pay. Not everyone will use this things these, but they are definitely not gimmicks. Samsung Pay is arguably better than both Android and Apple pay...
The Int'l version of the Galaxy S7 Edge is the world's top performing Android phone right now as well.
There are certainly some pros to the Axon 7 over the Galaxy phones though with the superior audio, from facing speakers, and higher built in storage to name a few. I'm just saying that it's not accurate to state that it's better at doing at a bunch of things or saying it's an overall better phone when it's not.
Btw regarding pricing, the Int'l version (superior version), has been continuously dropping each month with various sales. I think it's been hitting that 600 USD mark so it's looking like 200 USD premium.
It looks like the Axon is definitely the best bang for buck, but I can't call it a better phone than the S7 Edge.