How can I get this request to use the Custom Post Type page template instead?

Switch the taxonomy.php just like you would the single.php to support specific post custom templates. This is untested code, so it may not work – but, first get the post type

$post_type = get_query_var('post_type');

So if $post_type contains post_type=”car” then include archive-car.php like this:

if ( $post_type="car" ) 
  get_template_part('archive-car');
elseif ( $post_type="boat"  ) 
  get_template_part('archive-boat');
elseif ( $post_type="bike" ) 
  get_template_part('archive-bike');
else 
  get_template_part('regular-taxonomy');      

An better alternative would be if you could establish a template redirect like in this answer, but I don’t know if that’s even possible. If possible, you’d redirect the taxonomy template based on the value of $post_type.

How to quickly switch custom post type singular template?