**Core Concept**
The child's presentation with peripheral circulatory failure, along with the given arterial blood gas (ABG) values, suggests a state of metabolic acidosis with compensatory respiratory alkalosis. This condition is often seen in shock states, such as septic shock, where the body attempts to compensate for the metabolic acidosis by hyperventilating and blowing off CO2.

**Why the Correct Answer is Right**
In the setting of metabolic acidosis with compensatory respiratory alkalosis, the goal of immediate therapy is to support the patient's circulatory system and address the underlying cause of the acidosis. The correct answer involves the administration of a vasopressor to increase peripheral resistance and maintain blood pressure, thereby improving tissue perfusion. This is critical in shock states, where the body's ability to maintain blood flow to vital organs is compromised.

**Why Each Wrong Option is Incorrect**
* **Option A:** Administering insulin would be incorrect as the primary immediate therapy for peripheral circulatory failure in this scenario. While insulin may be useful in managing hyperglycemia and its effects on acid-base balance, it does not address the immediate need to support the circulatory system.
* **Option B:** Administering sodium bicarbonate would be incorrect as the primary immediate therapy for peripheral circulatory failure in this scenario. Sodium bicarbonate may be useful in severe metabolic acidosis, but it does not address the immediate need to support the circulatory system and may even worsen the situation by increasing blood pressure and potentially leading to cardiac arrest.
* **Option C:** Administering a diuretic would be incorrect as the primary immediate therapy for peripheral circulatory failure in this scenario. Diuretics may be useful in managing fluid overload, but they do not address the immediate need to support the circulatory system and may even worsen the situation by further decreasing blood volume.

**Clinical Pearl / High-Yield Fact**
In shock states, the body's ability to maintain blood flow to vital organs is compromised. The immediate therapy should focus on supporting the circulatory system and addressing the underlying cause of the acidosis. A mnemonic to remember the priorities in shock management is "ABCDE": Airway, Breathing, Circulation, Disability, Exposure.

**Correct Answer: C. Administer a vasopressor to increase peripheral resistance and maintain blood pressure.**