Laser beam propagation through an integrated fiber-optical system including a miniature cavity filled with a liquid crystal (LC) is numerically simulated. Two different shapes of the cavity are considered: a transverse cylindrical hole and a gap between the parallel end faces of the optical fiber. In both cases, the director field distribution in the LC volume includes a linear singularity (disclination). The Maxwell equations for an anisotropic continuous medium are solved by the FDTD method. Nonlinear effects of beam interaction with the LC medium are ignored. The simulations provide the data on the intensity distribution and direction of the laser beam that passed through the microscopic LC volume. The portions of laser radiation lost due to scattering into the ambient medium and remaining inside the optical fiber are calculated. Based on the calculations, it may be concluded that a significant fraction of the beam energy in the case with the cavity shaped as a transverse hole is scattered owing to focusing and diffraction induced by surface curvature and the finite transverse size of the hole. Moreover, there are regions with elevated energy density in the optical fiber behind the hole, which may lead to fiber fracture under the ultimate load possible in the pulsed mode. In contrast to the system with a hole, the deflection of beam propagation from a straight line in the fiber-optical system with a gap is fairly small. It is noted that beam-LC interaction can lead to the emergence of new fiber-optical modes in the transmitted beam.