Welcome to BYC
Sorry to know that one of your mallards has died. I realise that is sad for you. Although you do not mention how old your mallard was, from the description of malaise and not eating, she was clearly unwell. Those are very general, non-specific symptoms. There is nothing to suggest the cause of death.
My son has pekins and one of his died unexpectedly this week: she sat down by her swimming pool with the flock but didn't get up, she passed away sat there. We also don't know the cause. So I understand your anxieties.
However, I do not think you should treat your flock with an antibiotic when there is no evidence that the cause of death could have been prevented by using an antibiotic. Over use of veterinary antibiotics is a big reason humans have antibiotic resistant infections.
I advise that you only treat known infectiins with antibiotics