In a manner characterized by croup or croupy symptoms; in a way resembling the cough or condition of croup.
From croupy (adjective) + -ly (adverb suffix). Croupy itself comes from croup (respiratory disease) with the adjective-forming -y suffix.
Medical adverbs are hilariously specific—'croupily' means 'in that particular barking-seal way that croup has,' which parents hearing it at 2 AM would definitely recognize!
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