I funnily had the same use case. Two different jellyfin servers for complete separation. Both routing through gluetun. The reason this doesn't work is because the network mode setting you have basically makes all three containers operate in the same network. Meaning if one binds a port the others can no longer bind the same port. Their different hosts but all sharing one network and port range. To expose the ports you can move that ports setting from C1/C2 to the gluetun service definition. This'll still work because when C1 binds to 1234 it'll be reachable through the gluetun service.

Note: as mentioned if C1 and C2 cannot use the same port if you also want to have service gluetun set. More likely than not you start C1, it binds to the port, start C2, it tries and fails to bind to the port and crashes. I fixed this by making one of my jellyfin containers use a separate port. If you can't configure the ports of your services then there's no real recourse FWIU.