We propose a novel remapping approach tailored for multi-material Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics. This method is well-suited for integration with ALE frameworks capable of performing significant purely Lagrangian displacements between consecutive remesh/remap steps. Notably, it enforces pointwise constraints ensuring that the sum of all material volume fractions equals one. Given the initial and optimized meshes, we perform a direct interpolation of all material quantities in physical space. This bypasses the concept of PDE-based pseudotime advection. This step is computationally feasible due to recent advancements in parallel, GPU-capable interpolation routines provided by the GSLIB library. The resulting interpolation ensures bounded fields for each material with minimal diffusion but violates conservation. To restore conservation while preserving physical bounds, we employ two alternative optimization techniques. The first leverages interior point methods implemented via the HiOp optimization library. The second technique is based on the Proximal Galerkin Methods, performing optimization in a latent space that naturally enforces bound constraints. The proposed method produces minimal diffusion for finite element fields of arbitrary order and enables the direct remapping of quantities defined at integration points, eliminating the need to project onto finite element spaces. It circumvents challenges typically associated with synchronization between primal variables and conserved quantities in advection-based remap methods. We show results on standard 2D and 3D remap benchmarks and full ALE hydrodynamics simulations. This work is performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 (LLNL-ABS-2002031).