Wordpress category page customisation, list custom posts/item types by child category

I am customising a theme for Wordpress. Within this theme there is a category page template. I'm trying to get the category page to list posts by child categories.

Currently the code I have lists all of the posts with the child category as part of the post rather than under a heading for each child category.

So this is what I want the category page to look like:

Category Name
Category Description

Child Category 1 heading
Child Category 1 post
Child Category 1 post

Child Category 2 heading
Child Category 2 post
Child Category 2 post

Child Category 3 heading
Child Category 3 post
Child Category 3 post

The page needs to show all child categories and I need the ability to limit how many posts show for each heading.

Please see example files attached as it's slightly different to the usual way categories are displayed in Wordpress as it uses custom item types.