The needed part here is publishing (via windows installer)
So that you can specify "alternate" or "user interface" names for the stencil/template files.
You could do that as well with Visio Solution Publishing tool from Visio SDK (will require a bit more efforts though).
But yep, installer might not quite fit in the SVN "xcopy" deployment scenario - you'll need an extra step to pack stencils/templates in the installer somewhere in the process.
In principle, well, you can have a "build server" somewhere which pulls the templates/stencils out of the SVN and builds the installer for "end-users".
Or you could do that packaging via SVN post-commit hook for example. Not sure if it all worth the efforts though.
As far as I understand, you have a corporate LAN and cannot move your files outside of the company.. right?