Sharp geometric features play a central role in fluid--structure interaction (FSI), as nonsmooth boundaries can induce localized flow singularities. In incompressible Navier--Stokes flows, corners are particularly relevant, affecting the regularity of velocity gradients, pressure, vorticity, and viscous stresses. This effect is particularly relevant in immersed FSI, where the exchange of velocity and force information is mediated by transfer operators. When these operators rely only on averages over cells, elements, or boundary segments, localized corner contributions may be diffused and their strongly local character attenuated. To target this loss of localization, we propose a transfer operator based on a feature-aware quadrature of the mortar coupling matrices. The fluid-to-solid map is obtained through an L2 quasi-projection with lumped normalization, while both coupling matrices are assembled on a composite interaction set made of solid-volume samples, boundary samples, and explicit point samples at corners. After normalization, internal nodes are informed by volume samples only, boundary nodes by volume and boundary samples, and corner nodes by all three contributions. Thus lower dimensional entities receive increased weight in the coupling while the original nonmatching meshes and discrete spaces are left unchanged. The transpose map is then used for conservative force spreading. Numerically, the method provides a global response comparable to that obtained with standard immersed transfer operators, while improving the representation of corner-local quantities. In cases dominated by sharp corners and tips, the proposed strategy preserves sharper structural transients and more localized vorticity than purely volumetric coupling, suggesting improved sharp-feature FSI without changing meshes or multiplier spaces.