See, I don’t get that at all.
VLANs are different than what you have drawn.
A VLAN is just a virtual LAN. A VLAN is not a peice of cable.
A connection on a cable can contain one or more VLANs. If it has more than one VLAN, one or more of the VLAN’s pakets must be tagged with a a “dot1q” header, or encapsulated in some other way to keep the VLANs distinct… and the devices on both ends must understand those VLAN tags.
Having no knowledge of how this “BBOX” is handling VLAN tagging, I can’t tell you what should work and what shouldn’t.
… but you have emperically proven that you can’t connect the LiveWire twice to the same switch.