How to invoke MSBuild via command prompt?

From your comment, your web project is a web site project and not a web application project.

In this case, 'Publish' target can not be the option but 'AspNetCompiler' is the solution.

Create an xml file with below content and call it from MSBuild.

<Project xmlns="http://schemas.microsoft.com/developer/msbuild/2003">
    <Target Name="PrecompileWeb">
        <AspNetCompiler
            VirtualPath="/MyWebSite"
            PhysicalPath="c:\inetpub\wwwroot\MyWebSite\"
            TargetPath="c:\precompiledweb\MyWebSite\"
            Force="true"
            Debug="true"
            FixedNames="True"
        />
    </Target>
</Project>

Reference to this task is here and you can configure all your un/check options.

FixedName="True" equals the checking of 'use fixed naming and single page...' option.

Then you call this file from MSBuild instead of the solution file.

MSBuild your.xml /p:Configuration=<Debug/Release>

As long as your class libraries are referenced by your web site project, they'll be built together.

MSBuild.exe **MYPROJ**.sln 
/p:outdir="Z:\output\\",OutputPath="Z:\output\\",webprojectoutputdir="Z:\output\\",configuration=RELEASE 
/t:Clean;Build 

/flp1:logfile=Z:\output\\\Log_msbuild.log;verbosity=detailed 
/flp2:logfile=Z:\output\\\Log_warnings.log;warningsonly;verbosity=detailed 
/flp3:logfile=Z:\output\\\Log_errors.log;errorsonly;verbosity=detailed 

/nologo 
/noconlog

İt can compile all Types of projects(Web,webservice,desktop,..) and it can create log files as(buildlog,error and warnings).