Thank you for asking your query.

This particular coughing fits you are experiencing now could be related to the accidental exposure to the bleach and toilet cleaner mix. That said, it’s very difficult to say this is the sole reason for your trouble without physical examination and some tests. But it seems probable.
The gas which realized could have caused a server allergic reaction and might have ‘primed’ your lung to be hyper reactive to minor allergens. That could be the reason for this ‘cough fits’. Now coming to your specific question whether you should see your GP ? – Yes you should. He may refer you to a Pulmonologist if needed.

Its possible that each time when you get this little cough fits, there is microscopic damages happening which can accumulate over time and cause permanent lung damage.
And yes there are treatment options for this. Steroids under medical supervision can help this. Infections if any can be controlled by antibiotics.
Your Doctor may ask for an Xray or CT plus some blood tests to confirm the diagnosis.

Hope this helps