iPhone battery replacement

Tried this earlier, but did not get clean answers.

I have an iPhone6s whose battery lasts maybe an hour or so. In doing Battery Health, I get 96% "maximum capacity". Now, I'm told that should not be a sign that it needs replacement. But the work time I get a very bad. This battery has had a thousand or so charge cycles, so it is clearly old.

Is replacing the battery going to fix my problem? Sounds like it to me. But if so, why is Battery Health not telling me that?

FYI, the thing runs for many hours on a Mophie battery pack.