Show last child Child category instead of first child category on page

if I understand, you need to display category or taxonomy last child.
I think you should use get_terms function for this matter:
Get all first childrens from your taxonomy:

$my_childrens_firsts = get_terms( array('taxonomy' => 'your_taxonomy', 
                'hide_empty' => false, 
                'hierarchical' => true,
                'parent'=> $your_cat_id,
                ) );

Now you have to foreach $my_childrens_firsts (foreach($my_childrens_firsts as $key => $first_children)… so you get your $first_children->term_id

still within this foreach and get next childs level, using again get_terms:

$my_childrens_seconds = get_terms( array('taxonomy' => 'your_taxonomy', 
                'hide_empty' => false, 
                'hierarchical' => true,
                'parent'=> $first_children->term_id,
                ) );

and so on…until the sublevel you need.