I have seen this posted quite often on the Project Server forums so I thought I would write a quick blog post.
When upgrading from Project Server 2010 to Project Server 2013, you will need to recreate the customised Project Site templates. Project Server 2013 doesn’t recognise the Project Server 2010 project site templates. Project Server 2013 project sites now have a template name / ID of PROJECTSITE#0, Project Server 2010 project sites use PWS#0. This can be seen below:
2013:
2010:
When recreating the new site template in 2013, start with the “Project Site” template on the Collaboration tab found on the new SharePoint site page.
Modern Work Management - Project and the Power Platform blog 
Couldn’t you update the name and I’d using Set-SPSiteTemplate “PWS#0” –name “PROJECTSITE#0”?
Set-SPWebTemplate doesn’t exist in 2013. Also this command only has access to the installed web templates, not the templates deployed as a sandbox solution (custom templates). I haven’t yet found a way to get it working and always resorted to recreating the template. If I ever find or hear of a way to do this I will update this post.