Chat

Hi Gosia,
It is possibly true that your Moodle is unable to support that many user chats, maybe due to where it is hosted and what resources it has available. Before I retired, I self hosted Moodle on a server at home. I had to switch from a home account to a business account to get a large enough upload bandwidth to support the number of concurrent users I had in my lab when starting a quiz.
Have you considered having the tutors use some other free chat program? There are many available. This would let them use someone else's bandwidth.
Another possibility, have your tutors stream on a twitch.tv channel with their students watching the video stream and chatting in the chat window next to the video. It is a pretty robust system. I watched a live racing broadcast yesterday where there were so many people watching and chatting that the chat window was constantly scrolling, without a hiccup.