The problem seems to be that the default configuration added by the Common.Logging NuGet package (v2.0.0) is incorrect.
The runtime section in web.config needs to be changed to the following:
<dependentAssembly>
<assemblyIdentity name="Common.Logging" publicKeyToken="af08829b84f0328e" culture="neutral" />
<bindingRedirect oldVersion="0.0.0.0-65535.65535.65535.65535" newVersion="2.0.0.0" />
</dependentAssembly>
Note the oldVersion value. This seems to be what was causing the error (at least based on the scenario that I outlined in the question above).
See also this related GitHub issue: Possible Issues with Common.Logging / Common.Logging.NLog20 NuGet packages.