  We investigate the performance of a simple Bayesian fitting approach to
correct the cosmic microwave background (CMB) B-mode polarization for
gravitational lensing effects in the recovered probability distribution of the
tensor-to-scalar ratio. We perform a two-dimensional power spectrum fit of the
amplitude of the primordial B-modes (tensor-to-scalar ratio, $r$) and the
amplitude of the lensing B-modes (parameter $A_{lens}$), jointly with the
estimation of the astrophysical foregrounds including both synchrotron and
thermal dust emissions. Using this Bayesian framework, we forecast the ability
of the proposed CMB space mission LiteBIRD to constrain $r$ in the presence of
realistic lensing and foreground contributions. We compute the joint posterior
distribution of $r$ and $A_{lens}$, which we improve by adopting a prior on
$A_{lens}$ taken from the South Pole Telescope (SPT) measurement. As it applies
to the power spectrum, this approach cannot mitigate the uncertainty on $r$
that is due to E-mode cosmic variance transferred to B-modes by lensing, unlike
standard delensing techniques that are performed on maps. However, the method
allows to correct for the bias on $r$ induced by lensing, at the expense of a
larger uncertainty due to the increased volume of the parameter space. We
quantify, for different values of the tensor-to-scalar ratio, the trade-off
between bias correction and increase of uncertainty on $r$. For LiteBIRD
simulations, which include foregrounds and lensing contamination, we find that
correcting the foreground-cleaned CMB B-mode power spectrum for the lensing
bias, not the lensing cosmic variance, still guarantees a $3\sigma$ detection
of $r=5\times 10^{-3}$. The significance of the detection is increased to
$6\sigma$ when the current SPT prior on $A_{lens}$ is adopted.
