**Core Concept**
A 7-year-old child presenting with non-productive cough, mild stridor, and subsequent development of wheezing and productive cough, along with hyperlucency on X-ray and an obstructive pattern on pulmonary function tests (PFTs), suggests a condition affecting airway dynamics and potentially causing chronic inflammation.

**Why the Correct Answer is Right**
The child's initial presentation with non-productive cough and stridor, followed by an improvement with antibiotics, suggests a possible upper respiratory tract infection. However, the subsequent development of wheezing and productive cough, along with hyperlucency on X-ray and an obstructive pattern on PFTs, is indicative of airway obstruction. The combination of chronic symptoms and an obstructive pattern on PFTs points towards a diagnosis of **Bronchiolitis Obliterans Syndrome (BOS)**, also known as Swyer-James syndrome. BOS is characterized by chronic airflow obstruction, hyperlucency on chest X-ray, and an obstructive pattern on PFTs, often following a viral lower respiratory infection.

**Why Each Wrong Option is Incorrect**
* **Option A:** Cystic Fibrosis is a genetic disorder affecting the respiratory and gastrointestinal systems, but it typically presents with a more severe and early onset of symptoms, including malnutrition and failure to thrive.
* **Option B:** Asthma is a common condition affecting the airways, but it typically presents with episodic symptoms, reversible airflow obstruction, and a normal or hyperinflated chest X-ray.
* **Option C:** Bronchial foreign body aspiration can cause airway obstruction, but it is usually an acute presentation with sudden onset of symptoms, and the X-ray would show a foreign body in the airway.

**Clinical Pearl / High-Yield Fact**
In children presenting with chronic cough and wheezing, it's essential to consider the possibility of **Bronchiolitis Obliterans Syndrome (BOS)**, especially if there's a history of previous viral lower respiratory infections. A high index of suspicion and further diagnostic testing, such as PFTs and chest X-ray, can help establish the diagnosis.

**Correct Answer:** C.