Your employer is right. You have to get the card to start.

Based on my EAD, my card production order notice dated 9/10/07, after 5 days, they updated to send the card to my address, now my card shows 9/11/07 beginnng and 9/10/08 expired. You might tell your employer you need one week to receive the card because they have to see the card, and you can right now to prepare for work like volunteer these days. Dont take money and make trouble later. The offer letter doesnt need specific date. Like many H-1b, they need do transfer first after getting an offer letter. so Just tell them I can go to office to get familiar or training without pay first. just a thought.