If a person is entering on AP, there is a need for the person to show that he/she is currently working (having emp letter, paystubs is a good idea). H1 visa is not valid anymore since he has already traveled on AP. So there is a burden of proof with the traveler @ the POE to show work status and that can be only shown thru H1 or EAD.
Pls check with the lawyer for final advice. I think its risky to travel w/o a valid EAD.