Moisten the soil all around the root system of the bush and out beyond the branch area. This makes it much easier to dig out the roots.
Place your shovel at the outer edge of the branch area, and dig downward. Loosen the soil all the way around. Spiraling inward, use the pitchfork to gently lift and separate the roots. Do this completely around the bush, cutting a full circle. You will likely hear or feel roots breaking or snapping. If some roots are stubborn or too large, just clip them with loppers.
Dig under the plant with the shovel until you fully loosen all the soil and can lift out the bush. Leave it in the ground as you prepare a hole in the new location. This should be dug about twice the diameter and depth of the existing root ball you just dug out.
Lift the bush out of its original hole, and place it in the wheelbarrow to move farther away. Alternatively, if it's light enough or the new location is close by, just carry it to the new hole. Set the pomegranate bush into the new hole, and back-fill it with soil. Tamp down the soil firmly, and water it in well. Mulch around the bush with 2 or 3 inches of quality organic mulch or compost.