Gastrointestinal Bleeding due to Diverticulosis: This condition is a common cause of intermittent bright red blood in the stool, especially in older adults. The presence of conjunctival pallor, capillary refill of 4 seconds, tachycardia, and a systolic murmur suggests significant blood loss leading to hypovolemia and potentially anemia, which aligns with the symptoms of diverticulosis-induced bleeding.

Hemorrhoids: Although often associated with pain, hemorrhoids can cause bright red bleeding, especially if they are large or thrombosed. The other symptoms could be secondary to significant blood loss.
Anal Fissure: Typically painful, but in some cases, especially if large, can cause significant bleeding. The other symptoms could be related to blood loss and pain-induced tachycardia.
Inflammatory Bowel Disease (IBD): Conditions like ulcerative colitis or Crohn's disease can cause intermittent bright red blood in the stool, along with systemic symptoms due to inflammation and potential anemia.

Lower Gastrointestinal Malignancy: Colon cancer can present with intermittent bleeding, and it's crucial not to miss this diagnosis due to its significant implications on prognosis and treatment. The associated symptoms could be due to anemia from chronic blood loss.
Infectious Colitis: Certain infections can cause bloody diarrhea, and while less likely to present with the exact combination of symptoms described, missing this diagnosis could lead to severe consequences, especially in immunocompromised patients.
Ischemic Colitis: This condition can cause sudden onset of bloody diarrhea and is more common in older adults with vascular disease. It's a critical diagnosis not to miss due to the potential for bowel infarction and need for urgent intervention.

Angiodysplasia: A vascular anomaly of the gut that can cause recurrent, painless bleeding, often in older adults. It's less common but should be considered in the differential, especially if other causes are ruled out.
Meckel's Diverticulum: A congenital anomaly that can cause gastrointestinal bleeding, more commonly in children but can present in adults. It's a rare cause of bleeding but important to consider, especially if other diagnoses are excluded.
Arteriovenous Malformation (AVM): A rare cause of gastrointestinal bleeding that can present with intermittent bright red blood in the stool. AVMs can be congenital or acquired and may require specific diagnostic tests for identification.