This paper calculates the leading resistive accelerations acting on a system of two slightly deformed Reissner-Nordstr\"om singularities due to the emission of electromagnetic radiation, using matched asymptotic expansions. The unperturbed Reissner-Nordstr\"om solutions are assumed to have large charge-to-mass ratios $\frac{q}{m}$ and to be separated by a distance large compared to both $m$ and $\frac{{q}^{2}}{m}$. The problem is of interest primarily because of Rosenblum's use of point singularities in his calculation of the mechanical work done in small-angle gravitational scattering. Classical derivations of the electromagnetic equations of motion for a charged source were faced with the choice between indeterminate equations and divergences in the stress-energy. The use of asymptotic expansions about Reissner-Nordstr\"om solutions makes renormalization arguments unnecessary. The following paper compares the mechanical energy loss obtained from the present matching calculation to that predicted by the Lorentz-Dirac equation, which was derived using a point-particle assumption.