Final answer:
The likely pathogen causing the symptoms and presenting with a yellowish-gray thick membrane in the back of the throat is A. Corynebacterium diphtheriae, the causative agent of diphtheria.
Step-by-step explanation:
Based on the clinical presentation and findings such as a yellowish-gray thick membrane in the back of the throat which bled when scrapped, along with the results from the special cultures revealing Gram positive bacilli without spores, the likely pathogen is A. Corynebacterium diphtheriae. This bacterium is the causative agent of diphtheria, which is characterized by the formation of a pseudomembrane, high fever, and lethargy, especially in an unvaccinated child.