DON'T replace the battery in your 5S!! I'm sure it's fine, my 6 has 100% of its capacity still and I run into very similar battery issues. Games, high brightness, lots of photo capture/video recording, and cellular data cause atrocious battery life. Combine any of the two, and it automatically is safe to say that you will get less than 4-5 hours of use, which sucks. Again, this is what I've found on my 6, and everyone says the 6 has better battery life and is great. Yeah, I can say there's a difference but it isn't night and day.
I wouldn't say your idea is a bad one. The iPod can take the heavy load with games, while battery of your iPhone is preserved so it can serve its unique purposes, calls, SMS/MMS, and Internet usage on the go. The iPod would act as a good side kick, and as long as you have the funds, go for it.
If you would be breaking the bank for this, I would advise for you to move on and get a portable battery pack that can charge your iPhone a time or two.