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Hi, i`m Richard, a Veterinarian from the UK. Hopefully I can help today.
Feline panleukopenia (commonly referred to as feline distemper) commonly causes signs of anemia, this would have shown up on their bloodwork so is unlikely to be the cause. It is not possible for the vaccine to cause this disease.
Did your vet send fecal samples away for culture? Bacteria such as campylobacter or salmonella can cause the signs you are describing and may not be evident on a microscopic examination. If your vet did not send a sample away, this would be a good next step in investigating the cause of the diarrhea.
The convenia could potentially be feective against salmonella, but it would likely have little effect against campylobacter. It all depends on the sensitivity of the bacteria and only lab cultures would show that.