  We study competing orders of spinless fermions in the triangular-lattice
Hubbard model with nearest-neighbor interaction. We calculate the effective,
momentum-resolved two-particle vertex in an unbiased way in terms of the
functional renormalization group method and compare two different schemes for
the momentum discretization, one based on dividing the Fermi surface into
patches and one based on a channel decomposition. We study attractive and
repulsive nearest-neighbor interaction and find a competition of pairing and
charge instabilities. In the attractive case, a Pomeranchuk instability occurs
at Van Hove filling and $f$-wave and $p$-wave pairing emerge when the filling
is reduced. In the repulsive case, we obtain a charge density wave at Van Hove
filling and extended $p$-wave pairing with reduced filling. The $p$-wave
pairing solution is doubly degenerate and can realize chiral $p+ip$
superconductivity with different Chern numbers in the ground state. We discuss
implications for strongly correlated spin-orbit coupled hexagonal electron
systems such as moir\'e heterostructures.
