This is an interesting question -- given the details of this particular case, it is impossible to determine the exact cause and effect for your symptoms. Here are some possible explanations.
You may be experiencing some saltwater aspiration while you are breathing and rapidly changing the depth of your dive. Since salt water is hypertonic -- in this case, loaded with sodium chloride or salt -- tracheobronchial irritation can occur. This may be the cause of your discomfort when taking a deep breath. It should subside relatively shortly after your dive.
It may also be due to an irritating substance in your air supply. Any type of contaminant or residue from solvents, lubricants or combustible materials can irritate the mucus membrane lining when inhaled. If you obtain your air from the same source each time you dive for lobsters, this may be possible; if you buy your air from a single source each and every time you dive, including your lobster diving, this solution is not as reasonable.
Hyperventilation is another possible irritant. Breathing dry, compressed gas for two to three hours a day can dry the airway (assuming your activity does require an increased respiratory effort). The mucus membrane in the airway is responsible for providing moisture to humidify the dry, compressed scuba gas. It is possible to breathe dry gas at an accelerated rate (while exercising) that outstrips the airway capacity to humidify the air -- which creates the irritation of the throat and airway. This condition should also readily resolve once scuba diving has ceased.
Your discomfort might be due to mild barotrauma to your airway and lung tissue. Usual indications range from air bubbles under the skin (subcutaneous emphysema), to a collapsed lung (pneumothorax), to arterial gas embolism (AGE). Barotrauma results from the overinflation of the lungs when the gas volume expands against the soft tissues.
This may occur when a diver unintentionally holds their breath while changing the depth of their dive. In this case, we don't know if breath-holding occurred while breathing scuba or what depth changes, if any, occurred when pursuing lobsters. Mild barotrauma, however, is not a completely satisfactory answer, since gas volume changes seem to be small. You may have some pre-existing condition that causes your discomfort if you increase your physical exertion underwater and breathe scuba. For your own benefit, I would suggest that you be evaluated by your personal physician the next time you have symptoms.