I assume that template-blog-full-width.php is a page template and that you'd like to use this page template as the template to be used on the homepage.
What you can do is to create a copy of the template-blog-full-width.php template and name the copy front-page.php.
That's about it. WordPress will always use front-page.php template when on the homepage/frontpage of your website.
And just a side note
You might want to implement your changes by creating a child theme first so that any updates to the parent theme won't remove your changes..