The deployment of a managed database connection pooler in this context is the most effective solution. It helps in absorbing connection spikes, thereby maintaining database performance without exhausting its connection limit. AWS provides a service specifically for this purpose, allowing applications to pool database connections and resulting in increased efficiency for managing bursts of connections and ensuring the health and responsiveness of the database. This solution stands out because it preserves compute resources and maintains high database availability without requiring changes to the application. Other options like creating a buffering layer with a queue or enhancing compute resources of the database instance address different aspects of the problem but do not provide a direct solution for managing connection bursts as efficiently.