Bright green urates are classic for clymadia.
But having extra bilirubin in first morning poop is normal because they hold the poop overnight . So they can be bright green normally . I do not evaluate first morning poop . I only evaluate fresh poops after first morning..older poops can start having color leach out from the fecal part to falsely stain the liquid part..
Clymadia is a weird tricky bacteria, and because it is systemic it can cause different symptoms. It likes to hide in liver and kidneys and go through a dormant period . It doesn't cause elevated liver enzymes typically. The liver is very regenerative.
I actually had 6 of my parrots get exposed by a 7th I took in , so I had 7 with clymadia at the same time. They all had different symptoms, one the only symptom was weight loss , 2 the only symptom was the green urates and soft stools, one nearly died of septicemia,with free fluid in abdomen, intermittent clear fluid from one nostril , diarrhea, polyuria, vomiting...2 others were very sick, polyuria green urates , occasionally sneezing, red eyes. Anyway what I'm trying to share is it presents many different ways..plus there can be a chronic low grade infection .
If your vet wants to test for it , I definitely suggest you do .
Can you share a picture of each of your birds eyes? My birds had darkening of the iris and one developed a red tinge . Birds can show systemic inflammation in the eyes and not have weeping or swelling..it's kinda tricky at first to recognize. Infact the reason I caught it was I have a blue eyed bird who iris turned redish brown !! Then I realized my Brown eyed bird, iris turned so dark you couldn't see the pupil . This doesn't happen with every clymadia infection. But I'm curious, so I hope you will share.. especially as this is a cockatiel, and they often do