With Tawk you can use shortcuts to instantly reply to people to look at the FAQ if their issue is displayed there, but you would need to be in the chat with the person to do it.
You can also have private guides that only your staff can see on how to handle situations.
Upon checking the chatbox you linked, I noticed you can setup auto announcements. You could make an announcement every so often to remind people check the FAQ with a link to it, etc.
I think you can also make it visible on the front page on the forum as well so all members can see it, then do private messaging for one-on-one support.
If people do end up spamming help repeatedly you can give them a timeout or a ban, should the need arise. Then they could appeal it on the forum or a PM to you for a review.
I am also curious to know if the chatbox supports shortcuts, like, you can type in /faq in chat and it will display a message telling them to check the FAQ. In Tawk you can, you can also set up triggers so when a person sends in a message, it will remind them to check the FAQ and if the issue is not listed, to reply back to the chat. I haven't personally tried the triggers yet. I attached a screenshot.
I suggest giving both a try and see which one you like. Each have their own advantages and disadvantages, one is free, the other is paid. You could even use both at same time.