The p2g_type custom parameter can be used to distinct between ondemand, live scheduled and onair broadcasting.
To show a list of live scheduled events where the LTI group has view rights you can setup a external tool with the following custom parameter
To make this work you need to invite the group you used for LTI keypair to the scheduled event.
In addition you can combine this with e.g. p2g_search_restrictions=context_id or p2g_search_restrictions=context_title and populate the course_id or course_title in the scheduled event.
This forces to only show scheduled events meant for the specific course.
This selection will only show currently live broadcasting live channels where the LTI group was added to the permissions.
You can complement this with e.g. p2g_type=context_id. Make sure you populate the LTI Context ID metadata property with the correct context_id while broadcasting.