I struggled with this for a couple more hours and then went to plan 'B'
What I did was to get the cookie value in my initial script and pass it, via a post, to the LWP'd script. I modified my authenication subroutine (saved in a module), to accept first a cookie and, if that failed, get the cookie value from the posted data.
This works well and I do not have to worry about passing cookies. It also has an added bonus. I am now able to run my scripts if cookies are disabled by saving the cookie value in a hidden field.
Some day I'll understand how this cookie_jar thing works :-).