In some states, motorists under 21 receive a driver's license that is oriented as a "portrait" (long side vertical); those over 21 have a "landscape" license (long side horizontal).
Exhibit A Exhibit B
Why not simply run background checks on every Tom, Dick and Mary when they renew their licenses? Insta-check everyone. The "portrait" license or ID card would state "Possession of Firearms Prohibited" or something similar. Portrait indicates that you may NOT own a gun. Landscape shows you may.
Then if someone wants to purchase a firearm at a dealer or through a private transaction the transferor would be obligated to verify the purchaser was not prohibited from possessing a firearm. For dealers, this would replace the current Insta-check system and the DL would serve as proof of eligibility. No records for private sales, simple verification of eligibility. Licenses and ID cards would be good for three years before renewal.
What would ensure the integrity of the system? We the People, yo!