Not directly, no. Realize that your rectum passes directly above your prostate gland, so a passing bowel movement that is large can certainly press on your prostate. Since you have been to a urologist and already had a DRE (digital rectal exam) they would have been able to tell you if it were enlarged.
Keep in mind that a person does NOT have to have ALL the symptoms of a condition to have a condition.
There are any number of reasons you might feel a sharp pain in your urethra. Any small tear or small skin tag could be regularly irritated when the flow of urine passes by, kind of like air catching a kite. Since you can't stop urinating, the pain can last for a long time as the area keeps getting irritated over and over before fully healing. One might have expected the cytoscopy to see such a thing, but it's possible to miss it, and certainly not pleasant in and of itself.
I have no explanation why there might be a spike in sensitivity when you take a stool softener. When you have a strange reaction like that, you should contact the drug manufacturer, not look for listed complaints. They are going to have much more information than you will ever find publicly.
The other thing I can say is that internal pain is often misread by your brain as to its location. So while you feel the symptom in your urinary tract, the cause may be a nerve that is in your pelvis, but not actually in any way connected to your urinary system. The problem is that internal pain only has two settings, normal and excruciating. Once you have excruciating pain, the signal is so "loud" it can be very difficult for your brain to interpret the correct location of the source. Once the "loud" signal hits the local nerve cluster in your pelvis, it is over-exciting nearby nerves and your brain can't really tell what the origin of the pain actually is. Your symptom may be completely false and something else is causing the pain, for example a painful polyp in your lower colon that is aggravated by your constipation, and your brain does not know how to register this location.
It's hard to say, but a layperson isn't going to be able to figure this one out for you. It's going to require persistence with your doctor(s).