An important physical process unique to neutral-ion systems is the charge-transfer reaction. Here, we present measurements of and models for charge-transfer processes between cotrapped ultracold Ca atoms and Yb ions under well-controlled conditions. The theoretical analysis suggests the existence of three reaction mechanisms when lasers from a magneto-optical trap and an additional catalyst laser are present. We show that the near-degeneracy of the excited Ca(P11)+Yb+(S2) and Ca+(S2)+Yb(D23) asymptotic limits leads to large charge-transfer rate coefficients that can be controlled by changing the frequency of the catalyst laser and the ion temperature. Our model agrees with experimental rate-coefficient measurements between 50 mK and 1 K, with and without the catalyst laser, using just a single free parameter.