Use MeshBase method to check replicated. Add header for replicated mesh

[pshriwise]

Description

Adding the replicated_mesh.h header to mesh.cpp for libMesh compilations. Using the virtual
method for a replicated mesh check in one location as well.

While I wasn't able to reproduce the compilation error in #3687 it's good practice to include the header and not rely on a transitive include.

Fixes #3687

Checklist

• I have performed a self-review of my own code
• I have run clang-format (version 15) on any C++ source files (if applicable)
• I have followed the style guidelines for Python source files (if applicable)
• I have made corresponding changes to the documentation (if applicable)
• I have added tests that prove my fix is effective or that my feature works (if applicable)