  We study the nonlinear steady-state transport of spinless fermions through a
quantum dot with a local two-particle interaction. The dot degree of freedom is
in addition coupled to a phonon mode. This setup combines the nonequilibrium
physics of the interacting resonant level model and that of the
Anderson-Holstein model. The fermion-fermion interaction defies a perturbative
treatment. We mainly focus on the antiadiabatic limit, with the phonon
frequency being larger than the lead-dot tunneling rate. In this regime also
the fermion-boson coupling cannot be treated perturbatively. Our goal is
two-fold. We investigate the competing roles of the fermion-fermion and
fermion-boson interactions on the emergent low-energy scale $T_{\rm K}$ and
show how $T_{\rm K}$ manifests in the transport coefficients as well as the
current-voltage characteristics. For small to intermediate interactions, the
latter is in addition directly affected by both interactions independently.
With increasing fermion-boson interaction the Franck-Condon blockade suppresses
the current at small voltages and the emission of phonons leads to shoulders or
steps at multiples of the phonon frequency, while the local fermion-fermion
interaction implies a negative differential conductance at voltages larger than
$T_{\rm K}$. We, in addition, use the model to investigate the limitations of
our low-order truncated functional renormalization group approach on the
Keldysh contour. In particular, we quantify the role of the broken current
conservation.
