Similarily, Westboro Baptist Church protests are public safety issues, because they provoke large angry gatherings. How does the FBI handle that? Do they collaborate with corporations and local police departments to systematically discredit, propagandize against, and wage psychological warfare against the WBC?

Or do they do striaghtforward police work, provide a detail of officers at protests and counterprotests just in case anything gets out of hand, and generally behave reasonably?

There is a very far divide between "wholly ignorant" and "actively subverting." There is also a far divide between that and "sharing information on what to expect."

See also: undercover police excesses in the UK and US, including having children with activists to maintain cover.