Maybe the point was to stage a fake death of a 'patriot' and then have those who 'witnessed' the alleged shooting would go out into the crowd screaming, 'THEY (Law Enforcement) KILLED HER!' in hopes of firing up the crowd to become violent and attack Congressmen.
Only it didn't work because most people there weren't out for blood and they couldn't be driven to violence.
That certainly seems to be the case and it occurred to me too. But you can't fake a death and hide it forever.