Allergic Proctocolitis: This condition is characterized by the presence of blood and mucus in the stool of infants, often due to a milk protein allergy (most commonly to cow's milk). The fact that the infant is otherwise asymptomatic and gaining weight, and that the elimination diet has not completely resolved the issue, supports this diagnosis. The decrease in stool frequency could indicate a partial response to the diet, suggesting that while the allergen has been reduced, it may not have been completely eliminated or the infant may still be exposed to small amounts.

Infectious Gastroenteritis: Although the infant is asymptomatic aside from the bloody stools, a low-grade or resolved infection could still cause intermittent bleeding. The decrease in stool frequency might suggest the infection is resolving.
Toddler's Diarrhea: This condition is more common in slightly older infants but can present with mucus and occasional blood in the stool. The fact that the infant is gaining weight and otherwise well makes this a plausible option, though typically, it's associated with more frequent stools.
Food Protein-Induced Enterocolitis Syndrome (FPIES): This is an immune system reaction that occurs in response to certain foods, leading to inflammation in the colon, which can cause bloody stools. However, FPIES typically presents with more severe symptoms, including vomiting and diarrhea, which are not mentioned here.

Intussusception: Although rare in this age group without other symptoms like abdominal pain or a palpable mass, intussusception can cause bloody stools and must be considered due to its potential for serious complications if not promptly treated.
Hirschsprung's Disease: This congenital condition affects the colon and can cause severe constipation or obstipation, but in some cases, especially if the segment of affected colon is short, it might present with blood and mucus in the stool due to enterocolitis.
Inflammatory Bowel Disease (IBD): While rare in infants, conditions like Crohn's disease or ulcerative colitis can present with bloody stools. The absence of other symptoms like weight loss, failure to thrive, or abdominal pain makes this less likely but not impossible.

Juvenile Polyps: These are benign growths in the colon that can cause rectal bleeding. They are rare in infants but should be considered if other causes are ruled out.
Congenital Anomalies of the Colon: Certain anomalies, like colonic duplication or congenital vascular malformations, can cause bleeding. These are rare and usually present with other symptoms or findings.
Gastrointestinal Lymphoma or Other Tumors: Extremely rare in infants, but any persistent or unexplained gastrointestinal symptoms warrant a thorough investigation to rule out such serious conditions.