How do I render a field template?
In general, you don't access templates directly as an API, you build render arrays. To get the render array of a field, each field item and item list class has a view method. That method accepts either the name of a view mode and uses that configuration or you can also directly pass in the configuration you want to use (Basically the array that you have in view display config entities for each component).

You can try calling that from twig directly, e.g. {{ item.view('teaser') }} or prepare it in preprocess of your template.