yes give them a copy of the EAD. why would you want to pay at least $2000 to extend your H1? if you need to travel you get a AP (advanced parole) so they let you back in. right now in the eyes of the Immigration your status is Pending so just wait for your interview.
just a friendly advice, from your posting i see that your lawyer didnt explain H1 vs EAD. you need to sit down and talk to your lawyer about this and if he cant explain it to you, then find new lawyer because people who are in your shoes should be informed about conditions, pros and cons of the process.
