To solve problems regarding the interaction between a fluid flow and a structure, it is common to use unfitted mesh methods, where the discretised interface moves independently on a fixed fluid mesh. Among such methods, standard Fictitious-Domain (FD) ones are known to suffer from lack of mass conservation (see, e.g., [3] and the references therein). On the other hand, CutFEM and XFEM fix that issue and achieve optimal convergence, but they require an intricate treatment of the interface. Another possible approach consists of using discontinuous piecewise constant pressure, which requires high-order polynomials for the velocity. An exception to this limitation is the low-order divergence-free finite element pair introduced in [1] for the Stokes equations, where the P1 space for the velocity is enriched with one additional basis function per face, that is vector-valued, piecewise affine and has constant divergence on each simplex. This finite element pair has been shown to be stable and pressure-robust in the context of fitted meshes [1] and in an unfitted setting without an interface [2], in contrast to the instability of standard P1 − P0. In this work, we investigate this finite element approximation in the context of FD approximations of a Stokes problem with an immersed interface. To impose the immersed Dirichlet condition, we use standard P1 Lagrange multipliers, stabilised by either a Barbosa-Hughes or a Brezzi-Pitkäranta approach. Since the Lagrange multiplier is attached to a solid mesh that is unfitted to the fluid mesh, this introduces intricate quadratures to compute the term coupling the Lagrange multiplier and the velocity. We provide the analysis of the method and present the numerical results on well-known benchmarks, with particular emphasis on pressure robustness, on the two stabilisations and on the impact of inexact quadrature rules. Finally, the application to more general FSI problems will be numerically explored as well. [1] Erik Burman, Snorre H. Christiansen, and Peter Hansbo. “Application of a minimal compatible element to incompressible and nearly incompressible continuum mechanics”. [2] E. Burman, P. Hansbo, and Mats Larson. “Cut Finite Element Method for Divergence-Free Approximation of Incompressible Flow: A Lagrange Multiplier Approach”. [3] D. Corti, J. Diaz, M. Vidrascu, D. Chapelle, P. Moireau, et al.. "A fictitious domain method with enhanced interfacial mass conservation for immersed FSI with thin-walled solids".