iPhone turn off and then when you start it what is battery percentage. Does it dry fully ? I think you don’t need to replace battery.
secondly phone work for half day so I don’t think any hardware issue. Other wise phone would not work more than 3 to 5 minutes.
If you have updated any software then revert to old version.
And if you have not update software then restore iPhone with latest ios version. I am certain this is not hardware issue. it has some software related problem.