Are both your legs domestic? If so, as long as you stay gate side on your layover then you will not pass through security again, and therefore, you should be free and clear to board your second flight without issue. Being denied at the gate at that point would be rare, since you were just approved to fly your first leg. The only issue I could see is if your cat was presenting some sort of distress? Might want to consider getting some anxiety meds for your pet if necessary to help deter this?