Viral Gastroenteritis: This is the most likely diagnosis given the symptoms of loose green stool for 1 week and a recent onset of cough, which could be related to a viral infection affecting both the gastrointestinal and respiratory systems. The fact that the baby is breastfed and has moist mucous membranes with good capillary refill suggests adequate hydration, which is common in viral gastroenteritis.

Respiratory Syncytial Virus (RSV) Infection: Given the cough and the age of the baby, RSV infection is a possibility, especially during RSV season. However, the primary symptom of RSV is respiratory distress, which is not prominently featured here.
Acute Bronchiolitis: This could explain the cough and fussiness, especially if the baby has a viral infection. However, the green stool would need to be attributed to a separate cause or a viral infection affecting multiple systems.
Breast Milk Allergy or Intolerance: Though less common, an allergy or intolerance to breast milk (more accurately, to something in the mother's diet) could cause gastrointestinal symptoms. However, this would be unusual and not typically associated with a cough.

Bacterial Gastroenteritis: While less likely than viral causes, bacterial infections (e.g., Salmonella, E. coli) can cause severe diarrhea and need to be considered, especially if the baby shows signs of dehydration or if the stool is bloody.
Urinary Tract Infection (UTI): UTIs can present with nonspecific symptoms in infants, including fussiness and fever. A UTI would not directly cause green stool or cough but should be considered in the differential due to the potential severity if left untreated.
Pneumonia: If the cough is severe or accompanied by fever, pneumonia is a critical diagnosis not to miss. The presence of green stool might distract from respiratory symptoms, but pneumonia can be life-threatening and requires prompt diagnosis and treatment.

Congenital Anomalies: Certain congenital anomalies of the gastrointestinal tract could present with chronic or recurrent gastrointestinal symptoms. However, these would typically be identified earlier in life or have other associated symptoms.
Immunodeficiency: A condition affecting the immune system could predispose the baby to more severe or recurrent infections. This would be considered if the baby has a history of recurrent severe infections or fails to thrive.
Inflammatory Bowel Disease (IBD): Extremely rare in infants, IBD (Crohn's disease or ulcerative colitis) could cause chronic diarrhea and other gastrointestinal symptoms. However, this diagnosis is unlikely in a 5-month-old without a significant family history or other specific findings.