I have used iOpener on screen and it allowed to remove battery without damage to phone. Just heat the iOpener, place it on screen on side where the battery is, and leave it for a few minutes. Check with blue pick on bottom side if battery budges, if not repeat everything. I needed around 4 times to finally take out the battery.

Just finished battery replacement, worst part was taking out the battery. What helped me was putting iOpener on screen and checking every few minutes on battery if it will give up.
Also, battery from iFixit is not as good as I would think. Connector has a slight difference with original. Barely visible difference, yet plugging it was a challenge. I have brut forced it a little, but in the end it sits. But then another issue - cable is a little too long, which means that instead of lying flat, it bends a little. I had to squash it, I hope it will be fine for a time being. I am writing all this here because for some I can't give battery a review.
Cool from iFixit to provide fundamentals, but there are some details which unfortunately lay flat.