Maybe you could use the restrict access / user profile (name or surname or student ID) , or the personal messaging function instead of chat?
For the bookings you could use a padlet and paste the link on the moodle page. If you set the padlet so that students can 'write', then they can enter their names.
Here's an old Padlet example for I did for reserving Spanish conversation slots. https://padlet.com/shannonm5/oankgctci6ad