  We examine the thermal properties of the spin-1 Heisenberg antiferromagnet on
the honeycomb lattice in the presence of an easy-plane single-ion anisotropy as
well as the effects of an additional weak in-plane easy-axis anisotropy. In
particular, using large-scale quantum Monte Carlo simulations, we analyze the
scaling of the correlation length near the thermal phase transition into the
ordered phase. This allows us to quantify the temperature regime above the
critical point in which -- in spite of the additional in-plane easy-axis
anisotropy -- characteristic easy-plane physics, such as near a
Berezinskii-Kosterlitz-Thouless transition, can still be accessed. Our
theoretical analysis is motivated by recent neutron scattering studies of the
spin-1 compound BaNi${}_2$V${}_2$O${}_8$ in particular, and it addresses basic
quantum spin models for generic spin-1 systems with weak anisotropies, which we
probe over the full range of experimentally relevant correlation length scales.
