The Mean-Field Schrodinger Bridge (MFSB) problem is an optimization problem aiming to find the minimum effort control policy to drive a McKean-Vlassov stochastic differential equation from one probability measure to another. In the context of multi-agent control, the objective is to control the configuration of a swarm of identical, interacting cooperative agents, as captured by the time-varying probability measure of their state. Available methods for solving this problem for distributions with continuous support rely either on spatial discretizations of the problem's domain or on approximating optimal solutions using neural networks trained through stochastic optimization schemes. For agents following Linear Time Varying dynamics, and for Gaussian Mixture Model boundary distributions, we propose a highly efficient parameterization to approximate the optimal solutions of the corresponding MFSB in closed form, without any learning step. Our proposed approach consists of a mixture of elementary policies, each solving a Gaussian-to-Gaussian Covariance Steering problem from the components of the initial mixture to the components of the terminal mixture. Leveraging the semidefinite formulation of the Covariance Steering problem, the proposed solver can handle probabilistic constraints on the system's state while maintaining numerical tractability. We illustrate our approach on a variety of numerical examples.