I don't think the answer (for white people) to "race is a social construct" should be "go one-drop yourself into an ethnic history you've never experienced and are half making up on the spot." If my great-great-great grandma is from Spain but the last few generations of my family lived in Flyover USA, it'd be pretty weird for me to wake up one day and go around telling everyone I'm Spanish. Weirder still if that ancestor was from Catalonia and I try to say I'm personally affected by the Catalan independence movement.

A better response is to not actively identify with any race or ethnicity and try to do things that address the damage done by those concepts.