Two ways I have gotten batteries out of those tight boxes. One, I have a leather strap type handle with a steel end on each end that has a hole that fits over the terminals on top of the battery. Slide it down to thew bottom of the terminal and then as you lift it twists upward locking itself to each terminal and I can just lift the batter out. The other was was I had about an inch on each end of the battery and I got two steel strips about 1" wide and bent the bottom ends at right angles. After working them down beside the battery I could lift each end of the batter just enough to twist the steel strips and make the bent end go under the battery. Then I just lifted the two strips and they pulled the battery up. This is much easier on the hands of you tape or pad the top part of your lifting strips.
Ron