Here are my personal tips:

Engage haters, but with some rules. My first rule is always don’t get into a back and forth/debate with a hater. Give them one nice reply that addresses their statement and if they come back to argue. Just let it be. Hopefully sometimes they will come back to discuss politely and that can turn into a wonderful engagement. And always be honest in your response!

Be patient in engaging haters. Sometimes is best to let someone sit there for a bit, because another “fan” will respond and say something to the hater that you couldn’t have said.

Both of the above just take practice and time to get a feel for the people participating and what the right timing and amount of responses are.

If someone posts something critical, don’t delete it, acknowledge it. But if someone posts something disgusting and rude, I believe in deleting it. I haven’t worked with public figures. But at the company I worked for, most of the time I had that happen I would often message the person and explain why I deleted their comment or post. That also is a case by case.

Those are just a couple things off the top of my head. I hope they help out a bit. If you need more insight into them that I didn’t write, please message or post

Sounds like you are going to have an eventful Facebook page!