  We present the first detailed phenomenological analysis of a radiative
Majorana neutrino mass model constructed from opening up a $\Delta L = 2$
mass-dimension-11 effective operator constructed out of standard model fields.
While three such operators are generated, only one dominates neutrino mass
generation, namely $O_{47} = \overline{L^C} L \overline{Q^C} Q \overline{Q} Q^C
H H$, where $L$ denotes lepton doublet, $Q$ quark doublet and $H$ Higgs
doublet. The underlying renormalisable theory contains the scalars $S_1 \sim
(\bar{3},1,1/3)$ coupling as a diquark, $S_3 \sim (\bar{3},3,1/3)$ coupling as
a leptoquark, and $\Phi_3 \sim (3,3,2/3)$, which has no Yukawa couplings but
does couple to $S_1$ and $S_3$ in addition to the gauge fields. Neutrino masses
and mixings are generated at two-loop order. A feature of this model that is
different from many other radiative models is the lack of proportionality to
any quark and charged-lepton masses of the neutrino mass matrix. One
consequence is that the scale of new physics can be as high as $10^7$ TeV,
despite the operator having a high mass dimension. This raises the prospect
that $\Delta L = 2$ effective operators at even higher mass dimensions may,
when opened up, produce phenomenologically-viable radiative neutrino mass
models. The parameter space of the model is explored through benchmark slices
that are subject to experimental constraints from charged lepton
flavour-violating decays, rare meson decays and neutral-meson mixing. The
acceptable parameter space can accommodate the anomalies in $R_{K^{(*)}}$ and
the anomalous magnetic moment of the muon.
