CO2 levels follow warming because oceans emit CO2 as the warm and absorb it as they cool. The lag time is between 500 and 800 years.
You are correct about the historical lag of CO2 rise behind temperature rise. Thus CO2 increase has historically been a result of temperature increase, not a cause.
Unfortunately, the recent rise in anthropogenic CO2 caused by burning of fossil fuels is quite unprecedented, so past CO2 changes give us little information. This hasn’t happened before, so we cannot know what the effects might be.