Here's one demo suggestion:

    add_action( 'admin_enqueue_scripts', function()
    {
       wp_enqueue_script(    'my-script', '/my-script.js', ['underscore', 'backbone'], '1.0' );
       wp_add_inline_script( 'my-script', 'alert("hello world");' );

       // Add our template
       if( function_exists( 'wpse_add_inline_tmpl' ) )
           wpse_add_inline_tmpl( 
               $handle = 'my-script', 
               $id     = 'my-tmpl', 
               $tmpl   = '<div class="section intro">{{{ data.section.intro }}}</div>' 
           );
    
    } );

where we define our custom `wpse_add_inline_tmpl()` function as:

    function wpse_add_inline_tmpl( $handle, $id, $tmpl )
	{
        // Collect input data
		static $data = [];
		$data[$handle][$id] = $tmpl;

        // Append template for relevant script handle
		add_filter( 
            'script_loader_tag', 
            function( $tag, $hndl, $src ) use ( &$data, $handle, $id )
		    {	
                // Nothing to do if no match
			    if( ! isset( $data[$hndl][$id] ) )
                    return $tag;

                // Script tag replacement aka wp_add_inline_script()
      	        if ( false !== stripos( $data[$hndl][$id], '</script>' ) )
    			    $data[$hndl][$id] = trim( preg_replace( 
                        '#<script[^>]*>(.*)</script>#is', 
                        '$1', 
                         $data[$hndl][$id] 
                    ) );

                // Append template
				$tag .= sprintf(
				    "<script type='text/template' id='%s'>\n%s\n</script>" . PHP_EOL,
					esc_attr( $id ),
                    $data[$hndl][$id]
				);

			    return $tag;
		    }, 
       10, 3 );
	}
   
Here we use the `script_loader_tag` filter to inject the template code and use both the *handle* and *id* to reference it.  We also use the *script* tag [replacement][1] from the core `wp_add_inline_script()` function.

Hope you can test this further and modify to your needs. Writing this as a class might be a better approach ;-)


  [1]: https://github.com/WordPress/WordPress/blob/f645178391426ce859aad0bebf461516e1165123/wp-includes/functions.wp-scripts.php#L116