It might be the position of your body in the machine that’s causing you to have trouble breathing. Not all machines are created equal, and not every body type can fit into a machine. If you try to force your body to fit into one, you could end up compromising your posture that allows you to breath properly. This is akin to cyclists who have a harder time breathing when they use the aerobars on the bike to increase their aerodynamics; versus one who stay upright and can breath properly, but creates a lot of drag while riding.
Just because you use your core stabilizers, doesn’t mean that you can’t breath. Even core exercises that requires a lot of focus on stability will still give you the capacity to breath normally. If you are having trouble breathing while doing that type of exercise or any exercise that requires activation of the core stabilizers, then you must examine your form (or perhaps your own breathing pattern, such as instinctively holding your own breath while exercising).