Containerized deployment of microservices has gained immense traction across industries. To meet demand, traditional cloud providers offer container-as-a-service, where selection of the container and containerization of workloads remain developer’s responsibility. This task is arduous for a developer since the choice of containers across different cloud providers is many. Furthermore, there does not exist any mechanism using which one can compare and contrast the capabilities of containers across different providers. In this scenario, we envisage the need for a smart cloud broker that can automatically deploy a chosen IT service into the best-fit container environment mapped to performance requirements, from among the set of available underpinning brokered container hosting systems spread across multiple cloud providers. We propose a novel fitness-aware containerization-as-a-service to achieve this. We show why a best-fit container selection process is operationally complex and time consuming, and how we heuristically prune the associated decision tree in two phases so that it becomes viable to implement this as an on-demand service. We propose a new metric called fitness quotient ( <inline-formula><tex-math notation="LaTeX">$FQ$</tex-math></inline-formula> ) to evaluate containers obtained from heterogeneous providers. We leverage machine learning techniques to inject automation into these two phases: unsupervised K-Means clustering in the first-level build-time phase to accurately classify IaaS cost and performance data, and polynomial regression during the second-level provisioning-time phase to discover relationships between SaaS performance and container strength. We also show that the utility of the framework that we propose is not limited to the container fitness use case that we analyze in this paper; rather it can be generalized to address a class of problems where overall time and cost complexity for provisioning-time decision making needs to be controlled under a given set of constraints.