But I still wasn't exactly clear on all this. I can determine an
appropriate value for gamma to input into fix viscous. However, converting
this friction coefficient into a damping parameter for fix langevin is
unclear. I understand how a damping parameter in time units can act as an
effective force, and it is the same concept expressed in two different
ways. But, given a value for gamma, this friction coefficient in fix
viscous, how is the corresponding value for the damping parameter in fix
langevin determined?

It is done under the assumption, that the system is at the desired target temperature. It has to compensate the kinetic energy that is added.
If the instantaneous temperature is different, you thus get the thermostat functionality.

As the LAMMPS documentation kindly explains, the fix langevin
algorithm is based on

Dunweg and Paul, Int J of Modern Physics C, 2, 817-27 (1991).

Read that. Then consult a standard textbook account of Langevin
dynamics. Careful comparison of the Dunweg algorithm with the
theoretical underpinnings of Langevin dynamics will allow you to
directly relate the LAMMPS damp argument to the effective solvent
viscosity, at least for the limit of a non-interacting infinitely
dilute solute particle.