How to safely take screen of phone without breaking it

Did you review the guide here? iPhone 13 mini 屏幕更换

Are you sure the battery is the issue? If you have a Mac you can connect your iPhone to it to check its battery and charging using CoconutBattery Plus I find this very helpful.

There's always a risk that damage could happen when removing a screen, but iFixit guides are written to give you instructions in the safest way possible. Your phone's screen has a metal frame mounted on it, and by following the guide you'll be prying on the frame rather than the glass of the screen, minimizing the chance of breakage.

I've swapped out a lot of iPhone screens and haven't actually broken one yet, so not replacing your battery because you're afraid of breaking the screen isn't something to be worried about.