A kernel event listener may work for this. Render your mini template inside the listener and then "inject" it into the final response, similar to how the debug toolbar works:
Note that this is the approach you may want if you want to modify all pages universally. You can apply checks inside the listener to ignore ajax requests, redirection, and the like. You could also probably implement a simple map that knows which routes should have the response modified.
If you want to explicitly render the template in specific pages, then I agree with Maerlyn.