  A precise measurement of the top quark mass, a fundamental parameter of the
Standard Model, is among the most important goals of top quark studies at the
Large Hadron Collider. Apart from the standard methods, numerous new
observables and reconstruction techniques are employed to improve the overall
precision and to provide different sensitivities to various systematic
uncertainties. Recently, the normalised inverse invariant mass distribution of
the $t\bar{t}$ system and the leading extra jet not coming from the top quark
decays has been proposed for the $pp \to t\bar{t}j$ production process, denoted
as ${\cal R}(m_t^{pole},\rho_s)$. In this paper, a thorough study of different
theoretical predictions for this observable, however, with top quark decays
included, is carried out. We focus on fixed order NLO QCD calculations for the
di-lepton top quark decay channel at the LHC with $\sqrt{s}=13$ TeV. First, the
impact on the extraction of $m_t$ is investigated and afterwards the associated
uncertainties are quantified. In one approach we include all interferences,
off-shell effects and non-resonant backgrounds. This is contrasted with a
different approach with top quark decays in the narrow width approximation. In
the latter case, two cases are employed: NLO QCD corrections to the $pp\to
t\bar{t}j$ production process with leading order decays and the more
sophisticated case with QCD corrections and jet radiation present also in top
quark decays. The top quark mass sensitivity of ${\cal R}(m_t^{pole},\rho_s)$
is investigated and compared to other observables: the invariant mass of the
top anti-top pair, the minimal invariant mass of the $b$-jet and a charged
lepton as well as the total transverse momentum of the $t\bar{t}j$ system.
