Chronic Pulmonary Infection or Interstitial Lung Disease (ILD): Given the age of onset, symptoms of cough and fast breathing since two months, failure to thrive, hypoxemia, and the radiographic findings of diffuse ground glass opacities with subpleural sparing, this diagnosis is highly plausible. The lack of response to supplemental oxygen and the specific pattern on HRCT chest support a chronic condition affecting the lung parenchyma.

Pneumocystis jirovecii Pneumonia (PCP): Although the child's immune status is not explicitly mentioned, the presentation with hypoxemia, diffuse ground glass opacities on HRCT, and the absence of bacterial infection indicators (negative CRP, normal WBC differential) could suggest PCP, especially in an immunocompromised host. However, the lack of specific risk factors and the BAL findings do not strongly support this diagnosis.
Congenital or Acquired Immunodeficiency: The recurrent infections and failure to thrive could indicate an underlying immunodeficiency. However, the normal lymphocyte count and the absence of other infections make this less likely.
Allergic or Hypersensitivity Pneumonitis: Given the exposure to pigeons, this could be a consideration. However, the negative pigeon serum-specific IgG and the clinical presentation do not strongly support this diagnosis.

Cystic Fibrosis (CF): Although the sweat chloride test is normal (29 meq/L), CF should be considered due to the chronic respiratory symptoms and failure to thrive. Repeat testing or genetic analysis might be warranted if clinical suspicion remains high.
Tuberculosis (TB): Given the chronic nature of symptoms and the dirty proteinaceous background on BAL, TB should be considered, especially if the child is from a region with high TB prevalence. However, the lack of specific findings (e.g., granulomas on BAL) and no mention of TB contact history makes this less likely.
Congenital Heart Disease with Pulmonary Hypertension: Although the 2D ECHO is normal, pulmonary hypertension could explain the hypoxemia and failure to thrive. This diagnosis would require further evaluation, such as cardiac catheterization.

Lymphangiomatosis or Other Rare Lung Diseases: These conditions could present with chronic respiratory symptoms and diffuse lung involvement on imaging. However, they are rare and would require specific diagnostic tests (e.g., lung biopsy) for confirmation.
Surfactant Protein Deficiencies: These are rare genetic disorders leading to ILD in infants. The presentation and HRCT findings could be consistent, but specific genetic testing would be needed for diagnosis.
Leukocyte Adhesion Deficiency (LAD): This immunodeficiency could present with recurrent infections and failure to thrive. However, the normal WBC count and differential, along with the lack of specific infection history, make this less likely.