Need Correction
Eventhough DOB needs to be correct on both EAD & AP, I agree with your attorney as far as EAD is concerned. If the employer has any questions regarding the descrepancy, you may always produce your Driving Licence or any other official document to confirm your actual DOB and most of the employers would accept it as a mistake on the past of USCIS.
However, the incorrect DOB on AP needs immediate correction. When you will be producing the AP at POE, you would be dealing with an Immigration Inspector who would be unwilling to let you in on this flimsy ground even if you are able to show & he or she is willing to see the papers proving that it was USCIS which erred in giving you an AP with incorrect DOB.
Hence, act fast and send AP to the NSC for correction before you need to travel abroad.