  Electrons embedded in liquid $^3$He form mesoscopic bubbles with radii large
compared to the interatomic distance between $^3$He atoms, voids of
$N_{bubble}\approx 200$ $^3$He atoms, generating a negative ion with a large
effective mass that scatters thermal excitations. We develop scattering theory
of Bogoliubov quasiparticles by negative ions embedded in $^3$He-A that
incorporates the broken symmetries of $^3$He-A, particularly time-reversal and
mirror symmetry in a plane containing the chiral axis $\hat{\bf l}$. Multiple
scattering by the ion potential, combined with Andreev scattering by the chiral
order parameter, leads to a spectrum of Weyl Fermions bound to the ion that
support a mass current circulating the electron bubble - the mesoscopic
realization of chiral edge currents in superfluid $^3$He-A films. A consequence
is that electron bubbles embedded in $^3$He-A acquire angular momentum, ${\bf
L}\approx -(N_{bubble}/2)\hbar\,\hat{\bf l}$, inherited from the chiral ground
state. We extend the scattering theory to calculate the forces on a moving
electron bubble, both the Stokes drag and a transverse force, ${\bf F}_{W} =
\frac{e}{c}{\bf v}\times{\bf B}_{W}$, defined by an effective magnetic field,
${\bf B}_{W}\propto\hat{\bf l}$, generated by the scattering of thermal
quasiparticles off the spectrum of Weyl Fermions bound to the moving ion. The
transverse force is responsible for the anomalous Hall effect for electron
bubbles driven by an electric field reported by the RIKEN group. Our results
for the scattering cross section, drag and transverse forces on moving ions are
compared with experiments, and shown to provide a quantitative understanding of
the temperature dependence of the mobility and anomalous Hall angle for
electron bubbles in normal and superfluid $^3$He-A. We also discuss our results
in relation to earlier theoretical work on negative ions in superfluid $^3$He.
