Employer Overpayment

I was accidentally hugely overpaid during my internship in summer 2019 and this was reflected in my 2019 W2. I figured this out and filed an extension for my 2019 taxes. Now the company is saying that they cannot issue a new W2 for 2019 and I should pay taxes on the full income for 2019. They want me to pay them back the excess this year i.e. 2020 and use Pub 525 to get my excess 2019 tax back.
 
I have two questions:
1. Does this make sense? Is the company allowed to do this and not give me an updated W2?
2. I will be paying them a check this year. What documentation do I need from them to be able to use 525 in turbotax?

Thanks!