Is it possible to quit a TVL13 job in probation period?
If you want legal advise, then you need to talk to a lawyer or your union representative. It is a bad idea to trust random people on the internet on such issues.
My experience in Germany is that if you want to leave you don't go through that process. Instead you talk to your supervisor. (S)he will express regret and want to know why, but in the end (s)he will not want to keep someone who wants to leave. So together (with the HR department) you will make a contract that terminates the existing contract at a convenient time for both regardless of the notice periods.