iPhone disconnects from Wi-Fi when locked

I would back it up using icloud and another method like your computer.

You will need to do that if replacing hardware so practice is good.

Erase all data and settings in the General , Transfer or reset iphone navigation menu.

Set it up manually from Hello page with no backup restore. No installations . Turn off all icloud settings like photos. See if it still does it. If it does then hardware is quite likely.

All iPhones from the beginning of iPhone time 16 years ago disconnect from Wi-Fi 30 seconds after the phone screen locks, to preserve battery energy, unless the phone is connected to power. This is done because a Wi-Fi connection uses power continuously, whether data are transferring or not. Cellular connections use more energy, but only when actually sending data. There are a few exceptions:

• If you are streaming audio or video Wi-Fi will stay on
• If you receive a notification Wi-Fi will reconnect for a few seconds to process the notification
• And, I think (but I’m not sure) if an app uses Background App Refresh it can wake and use Wi-Fi; however very few apps actually use Background App Refresh.

Thete are a number of networking glitches including using VPN that can cause this

Try these remedies (except number 11)

https://www.fonegeek.com/ios-13/wifi-disconnects-when-iphone-is-locked.html

PS don't let it back up to iCloud in that state!

All 10?

All the remedies are software related and your latest additional information confirms this.