I haven’t used the Template plugin before, but I would guess that the template_templates.src property needs to be relative to the web root. So if your templates are in a public/htmlsnippets/ directory, you would pass e.g. 'src'=>'/htmlsnippets/name.html'.
Also judging by the docs for TinyMCE4, I’m not sure your config is valid. See TinyMCE | Template Plugin
Apologies for an absolute necropost however i had the need to implement this from a wordpress transfer to silverstripe. To get TinyMCE’s Templates/Snippet plugin working the config SetOptions should just be ‘templates’ not ‘template_templates’ for TinyMCE4.
or just reference a PHP file rather than an array and store your array as json HtmlEditorConfig::get('cms')->setOption('templates', 'tinytemplates.php');
with the following json
[
{"title": "HTMLSnippet", "description": "HTML Snippet", "content": "<p>My HTML content</p>"},
{"title": "HTMLTemplate", "description": "HTML Template from File", "url": "MyHTMLTemplate.html"}
]