Chronic Cough Related to Nerve Damage in Rectum

I had an accident with a sex toy in my anus about 5 months ago. It was painful but there was no blood and I didn't go to a hospital.

A few weeks after the incident, I developed a cough and since then I've had three chest infections.

I had an endoscopy done 10 days ago and the doctor confirmed some bruising about 4cm inside.

What I have noticed is that if I used hermorrhoid suppositories, it seems to alleviate the cough and chest feeling to a certain extent.

I also noticed that when pressure is applied to the tender area (like when the doctor checked it), my chest and cough felt worse the following day or two.

I now believe that I might have done some nerve damage (possibly due to an infection), in my rectum.

Is it possible that a damaged or exposed nerve in my rectum is causing me to have a cough and weird feelings in my chest?

I have seen a Pulmonologist and a Gastroenterologist and they both said that what I'm suggesting is not possible because the nerves for the chest/esophagus are too far apart from the nerves in the rectum. However, I strongly feel that the two things are linked.