Used a fake name for a cashback site and now PayPal wants ID to release my $40. What are my realistic options?

Hey everyone, I made a pretty foolish mistake and could use some advice.

I created a PayPal account using a fake name and address because I just wanted to redeem $40 from a cashback website. The money successfully transferred to the PayPal account, but now my account is limited and PayPal is asking me to provide a photo ID and proof of address to withdraw the funds.

Since the account information doesn't match my real identity at all, I obviously can't provide the documents they are asking for.

I'm not looking for shady hacks or illegal ways to bypass their security system, but I just want to know what my reality is right now over this $40.

1. Has anyone else been in this situation?

2)Is the money essentially gone forever?

3. Will PayPal eventually bounce the $40 back to the cashback site so I can claim it another way?

Any advice or harsh truths are welcome. Thanks in advance.