A completely white or clear discharge (not yellowish or creamy) that only comes with urination, or perhaps with bowel movements or other straining, probably reflects secretions from the prostate gland or seminal vesicles. The discharge of infectious urethritis sometimes is white or only slightly cloudy, but it occurs spontaneously, not just with urination or straining. And since you haven't been at risk for an STD, a non-infectious cause is even more likely. Such benign discharge can be entirely normal, and from what you say about your doctors' opinions, that's the case for you.
Your other symptoms are unrelated to your urinary symptoms. The "excess bacteria" you report, based on the way it was diagnosed (breath test) and the drugs used, was Helicobacter pylori, now known to be the main cause of peptic ulcer (which never was caused by stress or diet).
Bottom line: Sounds like your urologist and the other doc are on the right track, and that you have no STD or anything you need to worry about. But follow up with them if your symptoms continue or you remain concerned.
I have had a similar problem three times in the past year. I've found no connection to when it shows up and anything I've eaten or activity done etc. I too have kidney stones that have not caused any problems. I too had urine tests which showed nothing. I too saw my Dr. who could not find a cause. He further said he'd never heard of it and suggested it was left over semen however I've found no connection between when it shows and sex.
The one thing I have noticed is that it seems to come after my wife and I use a vibrator, particularly if the vibrator was on my lower stomach. Perhaps it's vibrating kidney stones?? When I suggested this to my Dr. he looked at me like I was nuts but I have seen this one correlation.