Classifying an infectious disease comes down to identifying the type of organism that causes it, and for leprosy, that organism has been known for well over a century.
Leprosy is caused by Mycobacterium leprae, a slow-growing, acid-fast bacillus that prefers cooler body tissues, which is why it mainly affects the skin and the peripheral nerves close to the body surface, causing the pale skin patches, sensory loss, and nerve thickening associated with the disease. Because a bacterium is the confirmed cause, leprosy sits firmly in the bacterial disease category, not among viral infections (which are caused by much smaller, non-cellular pathogens), fungal infections (caused by fungi such as dermatophytes or yeasts, which produce very different clinical signs), or metazoal diseases (caused by multicellular parasites like helminths, which is a completely separate class of organism from a bacillus).
Since none of viral, fungal, or metazoal fits the organism actually responsible, bacterial disease is the correct classification and the correct answer.