More than calcium I think you got it half right with the mulching. This is to keep water in. But if the tree is in a dry spot and free-draining that may not be enough. Additional watering as the fruits swell will help.
We had bitter pit when our apple tree was crowded by a neighbour's pear tree the other side of the fence. It was a real pity when they felled that tree, because half of the crown was leaning into our garden with nice pears, but as soon as the water and nutrient competition from the pear tree stopped, our bitter pit stopped too and we are now enjoying good apples.