The interaction of rod- or beam-like structures with three-dimensional continua (solids) can be found in a variety of different physical problems, for example in classical engineering applications (e.g., reinforced concrete and fiber-reinforced composites). In recent years, so-called mixed-dimensional formulations have emerged, where these problems are modeled by coupling 1D structural beam formulations to 3D structures. The main advantage of such formulations is that they allow for the representation of beam-like structures as 1D Cosserat continua. This allows for a highly flexible modeling approach for the coupled problem. For truly mixed-dimensional coupling between 1D beams and 3D solids--where the coupling constraints are directly defined on the 1D centerline of the beams--the resulting formulations are not asymptotically correct. This is due to the analytical solution exhibiting a singularity,cf. [1]. In many practical applications, truly mixed-dimensional coupling formulations can still be employed because the effect of this asymptotic inaccuracy only materializes for certain mesh size ratios--specifically, when the solid element size is smaller than the beam cross-section dimensions. However, an asymptotically correct coupling is highly desirable for problems where the beam cross-sections are larger than the solid elements. In [2], a method was introduced to resolve this issue using a Taylor series expansion of the coupling constraints around the beam centerline. However, this approach requires solid formulations with higher-order continuity, such as NURBS. In the proposed talk, we present an mortar-type approach to achieve an asymptotically correct coupling by defining the discrete Lagrange multiplier field on the surface of the beam to be in a Fourier finite-dimensional space, similar to the approach recently presented in [3]. Additionally, selected numerical examples will be presented to illustrate the proposed approach. [1] Steinbrecher, I., Mayr, M., Grill, M. J., Kremheller, J., Meier, C., and Popp, A., 2020, "A Mortar-Type Finite Element Approach for Embedding 1D Beams into 3D Solid Volumes," Computational Mechanics, 66(6), pp. 1377–1398. [2] Khristenko, U., Schuß, S., Krüger, M., Schmidt, F., Wohlmuth, B., and Hesch, C., 2021, "Multidimensional Coupling: A Variationally Consistent Approach to Fiber-Reinforced Materials," Computer Methods in Applied Mechanics and Engineering, 382, p. 113869. [3] Heltai, L., and Zunino, P., 2023, "Reduced L