You can achieve something like this with a combination of $site->visit() and $kirby->render().
Take a look at how the StaticBuilder plugin by @fvsch does it. I don’t know however if that would work inside another template as $site->visit() might break the state of the current page.
And I changed my mind. I had trouble to get the render function work with multi languages so I managed to use visit instead. I was inside a route so this worked (in the action):