As an attorney, we always remind clients to make copies of anything they send to USCIS (immigration) or any other government agency including proof of mailing. Unfortunately, if you did not keep a copy of the Employment Authorization Application (Form I-765) it will be difficult to know if the mistake on the EAD was something incorrect on the original application or a "Service error" (meaning a mistake by USCIS).
One way to view the application that was sent out is requesting a copy of your file from USCIS under the Freedom of Information Act (this is commonly known as a FOIA request). However, these requests can take months or over a year and probably will not be feasible for your situation.
Generally speaking, you should apply for a SS card under your correct last names especially if your existing identity documents (such as your birth certificate, passport) show your "correct" name.
To get an EAD card under your correct last names, you will have to send in a new EAD application requesting a replacement EAD and include a brief written explanation why you need a replacement due to the EAD card having your two last names switched. If this was a Service error, you would not have to pay the fee, but if not a Service error, you would have to pay the fee again.
I would highly suggest you contact an immigration attorney in your city or area who understands submitting replacement EAD applications to ensure any application is properly prepared and filed.
Good luck and best wishes.
The information is not legal advice and informational only. No attorney client relationship should be construed.
Make a new application for a work card please. Also send the bad card back to CIS. Please do not use a name that is not yours for purposes of convenience. This would offend CIS and any law enforcement officers who noticed what you did. Subtle changes in names suggest someone who is a fraudster. You do not want to identify yourself like that.
You may call the 800 number to see if they can suggest a solution, but I imagine they will say the same thing I did.
I am a pretty smart guy, but I do not know everything. My answers here come from the knowledge I carry around in my head--if you want me to look something up, you should come see me. Take any advice here as being general in nature. Please be careful with your life and make decisions that are smart.