Fortunately had some hours yesterday that I could spend. I fixed tons of issues in the client<->ice adapter communication and the connectivity check.
On joining a game with the new ice adapter hosted by a classic ice adapter, the classic ice adapter debug window shows a persistent connection that is kept being alive.
Unfortunately my game still does not join a lobby (stuck in joining lobby screen), so I guess there are some issues in the ice-adapter<->game connectivity that I need to figure out.