They do have reasonable evidence to at least suspect that you are an alt account of the account that has been previously banned, so from their perspective, they are simply trying to use the funds in your current account as 'motivation' for you to resolve any problems with the other account. Sort of like legal blackmail, if you will.
If someone has already encountered a similar problem, I ask lawyers to contact me to make a lawsuit against the illegal actions of the exchange.
You don't need any evidence that you own the account. You're trying to prove the other way around.
As squatter said, place emphasis on the fact that your documents were stolen and that there is $5k or so at stake. Don't go into the mumbo-jumbo of bitcoin jargon or too much about Localbitcoins, just keep it simple and ask them to file a case.