Using self-force methods, we consider the hyperbolic-type scattering of a
pointlike particle carrying a scalar charge $Q$ off a Schwarzschild black hole.
For given initial velocity and impact parameter, back-reaction from the scalar
field modifies the scattering angle by an amount $\propto {\textstyle \phantom{\rule{-0.167em}{0ex}}}{Q}^{2}$ , which we
calculate numerically for a large sample of orbits (neglecting the
gravitational self-force). Our results probe both strong-field and field-weak
scenarios, and in the latter case we find a good agreement with
post-Minkowskian expressions. The scalar-field self-force has a component
tangent to the four-velocity that exchanges particle's mass with scalar-field
energy, and we also compute this mass exchange as a function along the orbit.
The expressions we derive for the scattering angle (in terms of certain
integrals of the self-force along the orbit) can be used to obtain the
gravitational self-force correction to the angle in the physical problem of a
binary black hole with a large mass ratio. We discuss the remaining steps
necessary to achieve this goal.