We present high-precision ab initio calculations of the magnetic-dipole and electric-quadrupole hyperfine-structure constants for the $2p_{1/2}$ and $2p_{3/2}$ states of boron-like $^{35,37}$Cl$^{12+}$ ions. The calculations are carried out within the framework of bound-state quantum electrodynamics (QED) and include a systematic treatment of electron–electron correlation and radiative effects. The one-photon exchange and the one-loop self-energy and vacuum-polarization corrections are evaluated to first order in perturbation theory using established QED methods [Volotka et al., Phys. Rev. A 78, 062507 (2008)]. Higher-order contributions from interelectronic interactions are taken into account nonperturbatively within the Breit approximation by means of large-scale configuration–interaction calculations employing a Dirac–Fock–Sturm orbital basis [Tupitsyn, Opt. Spectrosc. 94, 319 (2003)]. Finite nuclear-size effects, including the Bohr–Weisskopf correction associated with the nuclear magnetization distribution, are incorporated within a single-particle nuclear model [Shabaev, J. Phys. B 27, 5825 (1994)]. A detailed analysis of theoretical uncertainties arising from uncalculated higher-order contributions is performed. The resulting theoretical predictions for the hyperfine splitting in boron-like chlorine ions are in excellent agreement with recent high-resolution spectroscopic measurements [Liu et al., Spectrochim. Acta Part B 235, 107349 (2026)], providing a stringent validation of the treatment of effects of interelectronic interactions and QED in mid-$Z$ highly charged ions.