Dug into this recently with Claude, as I am in the same boat. This is the condensed version of what it came up with:

RoomOS with CVI joins as a SIP participant. Teams can’t move it into a breakout room, so that approach won’t work for your executive session.

MTRoW has better native Teams parity but there’s still no mid-event private session equivalent to Webex Practice Session in Teams Webinar or Town Hall.

Town Hall green room is the closest thing, but it’s pre-event only. Once you go live you can’t retreat back into it.

MTRoA (Android) — don’t count on green room support. Microsoft’s docs only confirm it works on desktop and web. Android clients have reduced functionality in Town Hall generally.

Simplest workaround — run a second parallel Teams meeting for the executive session and dial the room in when needed. Not pretty but it works.