To manage outbound communication with a third-party API, the Ambassador container pattern is the most suitable choice. It acts as an outbound proxy that enables microservices to use an external service as if it was part of the local network, handling tasks such as authentication, monitoring, and circuit breaking. The Adapter container modifies or adapts the output of a container to a standardized format, but it is not primarily focused on outbound traffic management. The Proxy container is an oversimplified term that does not capture the specificity of the Ambassador's responsibilities. Lastly, while a Network bridge container is related to networking, it mainly refers to a container that builds network bridges, rather than managing external API communication.