  We analyse the phenomenological implications of a light Higgs boson, $h$,
within the CP-conserving 2-Higgs Doublet Model (2HDM) Type-I, for the detection
prospects of the charged $H^\pm$ state at Run II of the Large Hadron Collider
(LHC), assuming $\sqrt{s}=13$ TeV as energy and ${\cal O}(100~{\rm fb}^{-1})$
as luminosity. When sufficiently light, this $h$ state can open up the bosonic
decay channel $H^\pm \to W^{\pm(*)}h$, which may have a branching ratio
significantly exceeding those of the $H^\pm \to \tau\nu$ and $H^\pm \to cs$
channels. We perform a broad scan of the 2HDM Type-I parameter space, assuming
the heavier of the two CP-even Higgs bosons, $H$, to be the observed SM-like
state with a mass near 125 GeV. Through these scans we highlight regions in
which $m_{H^\pm} < m_t +m_b$ that are still consistent with the most recent
limits from experimental searches. We find in these regions that, when the
$H^\pm \to W^{\pm(*)}h$ decay mode is the dominant one, the $h$ can be highly
fermiophobic, with a considerably large decay rate in the $\gamma\gamma$
channel. This can result in the total cross section of the $\sigma(pp\to H^\pm
h \to W^{\pm(*)} + 4\gamma)$ process reaching up to ${\cal O}(100~{\rm fb})$.
We therefore investigate the possibility of observing this spectacular signal
at the LHC Run II.
