  Effect of the coupling to mobile carriers on the 120$^\circ$
antiferromagnetic state is investigated in a ferromagnetic Kondo lattice model
on a frustrated triangular lattice. Using a variational calculation for various
spin orderings up to a four-site unit cell, we identify the ground-state phase
diagram with focusing on the lightly-doped region. We find that an electron
doping from the band bottom immediately destabilizes a 120$^\circ$ coplanar
antiferromagnetic order and induces a noncoplanar three-sublattice ordering
accompanied by an intervening phase separation. This noncoplanar phase has an
umbrella-type spin configuration with a net magnetic moment and a finite spin
scalar chirality. This spin-canting state emerges in competition between the
antiferromagnetic superexchange interaction and the ferromagnetic
double-exchange interaction under geometrical frustration. In contrast, a hole
doping from the band top retains the 120$^\circ$-ordered state up to a finite
doping concentration and does not lead to a noncolpanar ordering.
