Obviously, You need to have battery changed as It is recommended by apple to change battery once battery goes below 80%.
The issue you are stating is not related to battery,
You should start from resetting network settings. (Disclaimer: You must need to know the wifi password.) Settings> General> Transfer or Reset> Reset > Reset Network Settings
Check for VPN or any Parental App or Security App. Remove if you have any. Check In Settings>General > VPN and Device Management.
Restart your phone
You should Make Back up of your phone, Connect your phone to the computer and update the phone or update on wifi in low power mode.
If issue still persist, Would recommend resetting all settings. (Disclaimer:It will make all settings to factory default. )
If issue still persist, Call apple Support or Go to Apple store or AASP have them to run diagnostics and they will get to know if there any hardware problem related to wifi hardware.