Can a Profile have hidden fields with preselescted or predefined values?
I've used a custom template page with jquery to do something similar. (I wanted the event to assume they were registering themselves and one other person - and NOT allow them to see they could choose how many.)

I wrote a quick jquery statement to preselect the value and then hide the option altogether with css.

https://civicrm.org/blog/hershel/how-customize-civicrm-pages-jquery