We present a phenomenology study on central exclusive production of $W^+W^-$ boson pairs in proton-proton collisions at the Large Hadron Collider at 14 TeV using the forward proton detectors, such as the ATLAS Forward Proton or the CMS-TOTEM Precision Proton Spectrometer detectors. Final states where at least one of the $W$ bosons decay hadronically in a large-radius jet are considered. The latter extends previous efforts that consider solely leptonic final states. A measurement of exclusive $W^+W^-$ also allows us to further constrain anomalous quartic gauge boson interactions between photons and $W$ bosons. Expected limits on anomalous quartic gauge couplings $a_{0,C}^W$ associated to dimension-six effective operators are derived for the hadronic, semi-leptonic, and leptonic final states. It is found that the couplings can be probed down to one-dimensional values of $a_{0}^W = 3.7times 10^{-7}$ GeV$^{-2}$ and $a_{C}^W = 9.2 times 10^{-7}$ GeV$^{-2}$ at $95%$ CL at an integrated luminosity of 300 fb$^{-1}$ by combining all final states, compared to values of about $a_{0}^W = 4times 10^{-6}$ GeV$^{-2}$ and $a_{C}^W = 1times 10^{-5}$ GeV$^{-2}$ at 95% CL expected for the leptonic channel alone.
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