I agree with the comment about pre-bending the cable first. I didn’t look close enough at the original battery and it looked like the replacement battery had a completely different cable. Then I realized that the cable turned up 180 degrees and then bent over 90 degrees.
That was fun!! I removed the volume side adhesive strip first. It broke immediately, but I was able to get it started again with the tweezers. It came right out. The inside strip was another story. It also broke immediately and I couldn’t get it started again with tweezers or anything. Since the volume side strip was out I used the black spudger tool to slowly start wedging the battery up and then I used a gift card to work the battery the rest of the way out. The gift card was not as thick as a credit card but was stiff enough to work the adhesive without bending the battery too much.
I decided to do the repair on my iPhone 6 without detaching the screen. I ended up screwing two pieces of 3/4” x 3” x 6” wood together at right angles. I used 2 rubber bands to secure the screen to the vertical piece and 1 rubber band to secure the base to the horizontal piece. One of the two adhesive strips broke and I had to really work to get the battery out. The rubber bands held everything together nicely. The wood pieces were slightly wider than the phone which made it easy to hold everything together while I was working the adhesive strip out.