  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 $a_{0}^W = 3.7\times 10^{-7}$ 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
$a_{0}^W = 4\times 10^{-6}$ GeV$^{-2}$ and $a_{C}^W = 1\times 10^{-5}$
GeV$^{-2}$ at 95\% CL expected for the leptonic channel alone.
