How to check during “pre_get_posts” if WP performing default query for specific custom template?

As you describe the timing would not quite work. During WP core load:

  1. main query is created and processed into query variables (with pre_get_posts being one of the hooks able to influence the results)
  2. then template-loader.php uses conditionals (dependent on those query variables) to determine template to use

So at the point of pre_get_posts you cannot technically check for template choice, because that is not happened yet. However what you can check for – is query variables are those that correspond with that template being chosen down the road. Which is practically quite close.