Step 1: Identify the danger signal: CNS depression and apnea in a baby after eye drops. This is a hallmark of central alpha-2 agonism.
Step 2: Brimonidine is that alpha-2 agonist used to cut intraocular pressure. Its problem is that an infant's blood-brain barrier is leaky, so the drug reaches the brain.
Step 3: Once central, it depresses the CNS and respiratory drive, producing bradycardia, hypothermia, hypotonia, hypotension and apnea. For this reason it is avoided in infants and young children.
Step 4: The beta-blocker, prostaglandin analogue and carbonic anhydrase inhibitor options do not carry this signature apnea risk, leaving brimonidine.\[\boxed{\text{Brimonidine}}\]