  We investigate the prospects of controlling charge-exchange in ultracold
collisions of heteroisotopic combinations of atoms and ions of the same
element. The treatment, readily applicable to alkali or alkanine-earth metals,
is illustrated in the process $^9$Be$^{+}$ + $^{10}$Be $\leftrightarrow$
$^{9}$Be + $^{10}$Be$^{+}$, which exhibits favorable electronic, nuclear, and
hyperfine structure. Feshbach resonances are obtained from quantum scattering
calculations in a standard coupled-channel formalism with non-BO terms
originating from the nuclear kinetic operator. Near a narrow resonance
predicted at 322 G, we find the charge-exchange rate coefficient to rise from
practically zero to values larger than $10^{-12}$ cm$^3$/s. Our results suggest
controllable charge-exchange reactions between different isotopes of suitable
atom-ion pairs with potential applications to quantum systems engineered to
study charge diffusion in trapped cold atom-ion mixtures and emulate many-body
physics.
