Took me awhile to get back to my originally posted question.
Final solution was to add this to the csproj file.
<Target Name="PublishToFileSystem" DependsOnTargets="PipelinePreDeployCopyAllFilesToOneFolder">
<Error Condition="'$(PublishDestination)'==''" Text="The PublishDestination property must be set to the intended publishing destination." />
<MakeDir Condition="!Exists($(PublishDestination))" Directories="$(PublishDirectories)" />
<ItemGroup>
<PublishFiles Include="$(_PackageTempDir)\**\*.*" />
</ItemGroup>
<Copy SourceFiles="@(PublishFiles)" DestinationFiles="@(PublishFiles->'$(PublishDestination)\%(RecursiveDir)%(Filename)%(Extension)')" SkipUnchangedFiles="true" />
</Target>
and then reference the Build Target: PublishToFileSystem
Parameters part of the MSBuild ended up looking like:
/p:Configuration=Release;PublishDestination=%WebSiteDirectory%
with %WebSiteDirectory%
being the destination location for the website.