Wet tail is just a diarrheal infection, which can be caused by parasites, bacterial , or other infectious disease colonizing the GI (due to inproper diet and malnutrition, or even poor husbandry).
If the infection is extensive enough it can ascend up the urinary tract and cause a secondary urinary tract infection.
You really need to go to a vet who has exotic pet knowledge and have your chinchilla examined, and have the appropriate tests performed, such as a culture and sensitivity of the stool, skin scraping of the affected area, parasite check and urinalysis, and whatever other tests the vet deems necessary.
There are also congential disorders that can cause the problem that you describe, and this can be diagnosed during the physical exam.
I'm not the vet, but I don't think you should wait for a response here before taking action. If the "place" you took the chinhilla was not a veterinary clinic, then that's where you need to go. If it was, you need to go to a different one that can handle exotic animals. With the presence of pus, he obviously has an infection that needs antibiotic treatment.