You cannot pick and choose who is entitled to receive constitutional rights. If you accept the Second Amendment is a right granted to all Americans by the constitution, then you can't tell some people they can't have that right just because they made a mistake when they were younger or suffered from a mental illness at one stage of their life.
Also, there is no indication to suggest that background checks would prevent mass shootings. Nicholas Cruz, the Parkland shooter, passed a background check to get his AR15. In fact, 80% of weapons used in mass shootings are obtained legally through the background check system.