According to your repro steps, I assumed that changing the output path in the application's property was your only change after you created the application. The only thing this change does is that it tells Visual Studio to put the output assemblies of MSBuild into the new folder. At runtime, however, ASP.Net wouldn't have any idea that it should load assemblies from this new folder instead of the \bin folder.
This answer shows the way to change the build output directory of a WebApi application. To get the exact same error showed in that post, you need to comment out the entire <system.codedom> section in web.config. And then you can follow the instructions to change the output path.
After you get your applicaiton work, you may then uncomment the <system.codedom> section. If you don't use C# 6 new syntax in your application at all you can uninstall the Microsoft.CodeDom.Providers.DotNetCompilerPlatform from you application; otherwise, you may want to add the following command line in your post-build event,
xcopy /Q /Y "$(TargetDir)roslyn\*.*" "$(TargetDir)..\roslyn\"
The new CodeDom provider always looks for the "\roslyn" folder in \bin. The above command works as a workaround and copies the \roslyn folder from your new output folder to \bin.
In my experiments, the publish tool of Visual Studio, however, published the output assemblies to the \bin folder in the deployment location regardless my output path setting. I guess your application should still work on actual deployment.