A super-massive black hole (SMBH) binary in the core of the blazar OJ 287 has been invoked in previous works to explain its observed optical flare quasi-periodicity. Following this picture, we investigate a hadronic origin for the X-ray and $gamma$-ray counterparts of the November 2015 major optical flare of this source. An impact outflow must result after the lighter SMBH (the secondary) crosses the accretion disc of the heavier one (the primary). We then consider acceleration of cosmic-ray (CR) protons in the shock driven by the impact outflow as it expands and collides with the active galactic nucleus (AGN) wind of the primary SMBH. We show that the emission of these CRs can reproduce the X-ray and $gamma$-ray flare data self-consistently with the optical component of the November 2015 major flare. The derived emission models are consistent with a magnetic field $B sim 5$ G in the emission region and a power-law index of $qsim2.2$ for the energy distribution of the emitting CRs. The mechanical luminosity of the AGN wind represents $lesssim 50%$ of the mass accretion power of the primary SMBH in all the derived emission profiles.
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