Can I use my own template for a view mode?
The standard way to do this is to create a template in your theme, you can use node--nodetype--viewmode.html.twig . (two dashes between each part).

To use it in your custom module, you'd have to use hook_theme_registry_alter to change the path that it looks in for that theme hook.

function mymodule_theme_registry_alter(&amp;$theme_registry) {
  $theme_registry['node__nodetype__viewmode']['path'] = drupal_get_path('module', 'mymodule') . '/templates/';
}


I don't really see the value of moving it into a custom module though.  It's a standard to keep custom templates in the theme.