The Hassle Free clinic is great, but any walk-in clinics will handle an STI.
The time from exposure to symptoms is the most helpful bit of information to tell them.
You can safely make the rest of the story up. When you explain to the doc why you are concerned, feel free to use the same level of truthfulness that you used to explain where you were that day. Easiest is to just say "the condom broke" and (describe symptoms) or if no symptoms, just say "I want to be sure that I didn't pick something up".
They'll like you for acting responsibly.
Bottom Line: The medical community would rather hear a bull-shit story that everyone pretends is true, than have someone feel too ashamed to come in to be tested and treated.
In your case, if you had stinging right after, then it was just chafing.
But, having the tip of your penis sting 2 days after is probably an infection, very typical of Gonorrhea, but could also be a non-STI infection, like a strep or E. coli.
Chlamydia usually takes a bit longer to develop and Syphilis takes a few weeks longer.
The problem is that the symptoms often settle down and you think it is gone, but it can still be there, just silent and spreadable.
So get tested even if you are feeling better!
The clinic will just need a urine test. And if some vigilante doc offers to stuff a swab up your urethra, politely decline.
Modern urine testing is just as good and a swab stuffed into an inflamed urethra, hurts like hell.
All testing is done by the Public Health Lab and Public Health provides clinics with free antibiotics to treat Chlamydia, Gonorrhea and Syphilis.
