So, I figured out how to achieve what I wanted to achieve and am outlining the steps below hoping that it might help someone else out.
First, to add an external URL to the assignment object, pass:
assignment[external_tool_tag_attributes][url] = 'https://my_favorite_url.com'
to your list of parameters. Your assignment will then contain the external tool URL.
The tricky part was then how to create custom assignments: I needed the external URL to differ by a custom field to help in specifying a specific assignment to help with grade pass back. My solution was to create files which had names like: https://my_favorite_site.com/assignments/1.php which were created at the time of the assignment creation.
Then, in that file I had something like:
$the_get_variable_I_want = basename(__FILE__, '.php');
require_once(DOCUMENT_ROOT . 'the_place_where_I_do_my_LTI_stuff.php');
Then, my LTI file will be passed $the_get_variable_I_want. So, I'll have one external tool url for each assignment and then grades can be passed back accordingly. Finally, for those of you who need multiple custom variables, the "1" above could represent a record in your database, saved at the time of assignment creation by assignment id, and then used to look up any number of variables that you actually need. Or, as another alternative, your file could be something like 1-2-3.php and then you can pull out the 1, 2, 3 and use them accordingly.
I hope that at least one of you will find this helpful.