Because of the way the database is structured it would involve editing the WeBWorK code to add additional columns to the Classlist. If you're not already using the Section and Recitation columns for registration information you could put the degree route information into one of those. Both of those are stored as text in the database, so you can put whatever information you want in them.
In terms of what information can be passed to WeBWorK from Blackboard, you can customize WeBWorK to accept anything that Blackboard sends. Take a look at the modify_user code in the LTI configuration file (the name of this file depends on which version of WW you are running). You can use that to take any parameter passed by Blackboard and store it in any column in the users table.