IANAL (I Am Not A Lawyer)
Legally I am pretty sure you can refuse to sign a false document.
It kind of depends on where the verification came from. A government agency it may be criminal to sign false information. If it is a company they might be able to sue for damages.
Your company could be mad if you sign a false document. Your boss may deny she forced you to sign the document.
For sure your reputation is hurt by signing a false document. It could get out in the workforce and community that you are not trustworthy.
If you just refuse to your boss she might retaliate and you have no proof she verbally told you to sign. She is not a person of integrity based on what has happened.
If you go up the chain (HR or her boss) she still could deny she told you to sign. But it just does not make sense that you would lie about this if you reported it up front. Even if they believe you and reprimand your boss you could still suffer collateral damage.
The fact you have a false document takes away from the credibility of your boss and her daughter.
Boss does not want to sign because of last name does not hold water. They are not going to question a validation based on a matching name. Your boss just does not want to sign a false document and is bullying you. I bet this validation was sent to you by name. Typically a validation would go to HR.
I do not think you should sign and deny validation. That is something that should be cleared by HR.
You are in a bad spot and should not sign in my opinion. If you just refuse to your boss or go to HR is a harder question.
It just does not make sense to lie about this in the first place. It is not likely to be the difference in getting a job and getting caught would be bad for everyone.