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We leverage the crucial hyperelastic properties of a multifunctional structured surface to optimize the reconfigurability of the electromagnetic transmission under large nonlinear mechanical deformations. This multiphysics, multifunctional, hyperelastic structured surface (HSS) offers two simultaneous intriguing functionalities; tunability and switchability. It is made of copper resonators and a Polydimethylsiloxane (PDMS) substrate, which is one of the most favorable deformable substrates due to its hyperelastic behavior. The proposed HSS is fabricated via an original cost-effective technique and the multiphysics functionalities are captured in both experimental tests and numerical simulations. Leveraging the hyperelastic behavior, we demonstrate up to 8% percent shift in the resonance frequency in the GHz range, for average applied mechanical strains of around 17%. The hyperelastic deformations can continuously increase/decrease the magnitude of the scattering parameter S21 in the frequency range of 10.9 GHz to 11.8 GHz by more than 40 dB, changing from being largely transparent to opaque and vice versa. The potential of hyperelastic behavior to account for the multifunctionality of the HSS is validated experimentally.