Mixed-dimensional finite element formulations that couple one-dimensional beam models with three-dimensional solids via Lagrange multipliers naturally lead to large, indefinite saddle-point systems. Their efficient solution is challenging due to strong stiffness contrasts, geometric scale separation, and the presence of a pure Neumann subproblem on the beam block. In this work, we propose a modified augmented Lagrangian block preconditioner for the coupling of torsion-free Kirchhoff–Love beams embedded in a three-dimensional elastic solid. The augmented formulation regularizes the coupled system while preserving exact enforcement of the coupling constraints, thereby improving the conditioning of the linear system and eliminating the singular character of the beam subproblem. Starting from an ideal augmented Lagrangian block preconditioner, we derive a practical block-triangular variant that enables independent and scalable treatment of the solid, beam, and Lagrange multiplier blocks. Algebraic multigrid is employed for the solid subproblem, while the remaining blocks are treated with efficient direct or approximate solvers. Various Schur complement approximations are introduced and analyzed, highlighting the role of the augmentation parameter and the trade-off between robustness and computational cost. Numerical experiments on representative volume elements of short fiber-reinforced materials demonstrate near mesh-independent iteration counts and robustness with respect to stiffness ratios, beam radii, mesh-size ratios, and fiber-volume fractions. Strong and weak scaling studies on distributed-memory architectures confirm the suitability of the proposed approach for large-scale simulations. The results show that the modified augmented Lagrangian preconditioner provides a robust and scalable solver strategy for mixed-dimensional beam-solid interaction problems arising in computational mechanics.