Within this contribution, a novel computational framework based a regularized mortar-type finite element discretization is presented, that allows to address unique challenges arising from mixed-dimensional coupling of very slender fibers embedded in fluid flow. The fibers are modeled via one-dimensional (1D) partial differential equations based on geometrically exact nonlinear beam theory, while the flow is described by the three-dimensional (3D) incompressible Navier-Stokes equations. The arising mixed-dimensional 1D-3D coupling scheme requires tailored solution schemes to ensure an accurate and efficient computational treatment [1,2]. In particular, we present a strongly coupled partitioned solution algorithm based on a quasi-Newton method for applications involving fibers with high slenderness ratios that usually present a challenge with regard to the well-known added mass effect. The influence of all employed algorithmic and numerical parameters, namely the acceleration technique, the constraint regularization parameter as well as interpolation functions, on efficiency and results of the solution procedure is studied through appropriate examples. The convergence of both one-way and two-way coupled mixed-dimensional solutions under uniform mesh refinement is demonstrated, a comparison to a 3D reference solution is performed, and the method's capabilities in capturing flow phenomena at large geometric scale separation is illustrated by the example of a submersed vegetation canopy. Aspects of domain partitioning and parallel computing for the resulting simulation models are highlighted alongside the numerical examples.