My advice would be to wait for a few months. In immigration matters, and actually in all legal matters, honesty is the best policy. The risks of telling a lie are huge in all immigration cases but they are particularly high in this one: there is no appeals process. So if the prosecutor has even a little bit of doubt he can reject your application and you would not be able to do anything about it. This is even worse than what rights someone has for a tourist visa.
Your case is a complex one for sure. You have a messy history with multiple SS# and then you have a taxes paper trail. The advantage of waiting are twofold. You can watch what happens to others and then respond accordingly. Two, you can simply decide that you would rather wait for an amnesty that will allow you to erase your past crimes. The last thing you want is for your application to be rejected and then also charged with identity theft and tax fraud.
In any case, I strongly encourage you to discuss this with your attorney but also think it through yourself because many attorneys simply are interested in you filing the application so that they can get paid. So just pay for an hour of consultation first.