The FBI says that it’s a felony to use the CDC’s logo without permission, and can net you a 5 year jail term and thousands in fines.
No one keeps a searchable record of who has and has not been vaccinated.
So an answer seems easy to me. Alter the logo, then make the alteration difficult to see. Damage and dog ear the fake card to the point that it looks well worn and the logo is smudged as to be nearly unidentifiable.