Elliptic interface problems play a central role in many scientific applications, such as fluid-structure interaction (FSI). A recent method, known as FD-DLM, couples unfitted finite element discretizations with the Fictitious Domain approach using Distributed Lagrange Multipliers. Although effective in handling complex geometries, this approach results in large, ill-conditioned linear systems that are computationally expensive in terms of both time and memory. To address this issue, we focus on designing robust preconditioners tailored to elliptic interface problems within the FD-DLM framework. However, further challenges arise: in addition to maintaining stable iteration counts across mesh refinements, a key and highly desirable feature of a preconditioner is robustness with respect to potentially large jumps in the coefficients. Furthermore, the design of block preconditioners must consider the presence of a singular (2,2)-block, due to the lack of boundary conditions on the immersed domain. We tackle these challenges by extending to the FD-DLM framework the augmented Lagrangian (AL) strategy that we proposed for boundary-supported Lagrange multipliers. Specifically, we introduce a modified AL block-triangular preconditioner that simplifies implementation while maintaining low iteration counts. We evaluate our method using FGMRES on 2D and 3D numerical experiments conducted with the C++ finite element library DEAL.II, demonstrating robust performance with respect to both mesh refinement and large coefficient jumps.