Update: I received the suggestion to run the client through the command prompt, which output the following error upon build:
ERROR: godoticallBuildInstancegetMSBuildPath: Cannot find executable for 'MSBuild (VS Build Tools)'. Tried with path:
System.IO.FileNotFoundException: Cannot find the MSBuild executable.
at GodotSharpTools.Build.BuildInstance.GetMSBuildPath () [0x00013] in <32a76a9df224459d83b0375c3bbf8619>:0
at GodotSharpTools.Build.BuildInstance.Build (System.String loggerAssemblyPath, System.String loggerOutputDir, System.String customProperties) [0x0001a] in <32a76a9df224459d83b0375c3bbf8619>:0
Upon further Googling, it seems that others have had similar issues related to the path where MSBuild tools are supposed to live. I went ahead and uninstalled Visual Studio 2017 Community and installed 2019. That seemed to work. [shrug]