Lying about being a LPR does not carry the same type of penalties as lying about citizenship, and it may or may not have negative consequences.

If you say that you are a permanent resident, they could in theory get you for trying to defraud a financial institution (which is a federal crime with hefty penalties as all commercial banks are insured by FDIC) BUT they will have to prove your intent was to defraud them.

At the same time, lying and fraud are not the same thing. In order for someone to be found guilty of fraud, the victim has to prove that the defendant's actions resulted in clear and predictable harm to the plaintiff, and that the result was also their main intention all along. In other words, lying about your status to get a credit card that you expect to use responsibly does not necessarily constitute fraud, but if you plan to get 10 cards and simply max them out and default later on(this actually being perfectly legal in itself), the lie could constitute enough proof for fraud.