How automatically enter the “meta_data_filter_cat” when an user inserts a new article

Meta filter category ID  - keeps in each post with the same name key 'meta_data_filter_cat'.

So if you want to allow your user create new posts with already predefined or selected 'Meta filter category ID' you are need in your wp theme functions.php  add next code:

Where $meta_data_filter_cat_id is predefined meta_data_filter_cat ID which of course you can get using array $_POST if you are allowing your user to select filter category filter ID on the front ...