Using numerical methods for solving partial differential equations, we often want our approximations to fulfill continuity equations associated with the PDE, such as mass conservation. We present a conservative Cut Finite Element Method (CutFEM) for solving a convection-diffusion equation in an evolving domain simulating the transport of surfactant concentration in immiscible fluids. Discrete mass conservation is achieved up to machine precision by leveraging Reynold's transport theorem. We introduce a time-dependent macroelement stabilization for space-time CutFEM discretizations that reduces the error, increases system matrix sparsity, and decreases the sensitivity of the error with respect to the stabilization parameter, making it a compelling development of previous work [1]. We obtain a family of unfitted high-order space-time discretizations that have optimal convergence rates, maintain the scaling of the system matrix’s condition number with respect to mesh size from fitted finite element methods, and conserve the surfactant mass.