thank you for your question. So first of all after your Ausbildung you have the legal right to work 2 yeas and be not deported( does your Ausbildung Comapny offer you a job after the the Ausbildung?). If threy cannot take you after the Ausbildung, please start looking for a company that might be interested to take you straight after June.
As for the wrong details please share it with your lawyer as he/she has all the relevant information and is able to write the authorities.
Tell the lawyer as you outlined here that you listened to the wrong information given before you applied for your asylum, you might have to hand in proof of your real identity.
Do not get scared or panic! If you need further assistance please do not hesitate to reach out to us.