Magnetic skyrmion crystals are hexagonal lattices whose sites are occupied by individually deformable, topologically protected quasiparticles. Unlike atomic crystals, skyrmions can continuously reshape, raising a fundamental question of how such a lattice accommodates discrete defects. Using phase-field simulations, we reveal that skyrmion deformability regularizes disclinations into finite cores as core skyrmions deform and split into half-skyrmion pairs. Within this finite core, continuous skyrmion deformation accommodates the skyrmion-lattice strain that would otherwise diverge. We extract effective core radii from the radial profiles of particle strain, hexatic orientational order, and enclosed Frank angle; these three measures define distinct length scales that broaden with increasing skyrmion deformability, revealing a multi-scale core architecture. The inter-core skyrmion-lattice strain is reduced relative to the point-core Volterra quadrupole reference. These results demonstrate the coexistence of continuous constituent deformability and discrete lattice topology and elucidate how their interplay shapes the physics of disclinations in skyrmion crystals.