Normally you can use a Beomedia 1 with a avant VCR.
Instead of the PC command (which the avant most likely doesnt know), you can change the setting from PC to VAux in the setup menu from the BM1.
The videosignal can be connected through composite (1 yellow lead) on one of the scarts or via the doorcam/camera input.
The problem your facing is the lack of a free scart.
Depending on the software of the avant, you can change in the setup of the avant the doorcam/camera instruction to another instructie, eg. VAux, Doorcam etc. But that depends on how old and which type the avant is. As it seems to be an RF version, there is software available to make it possible.
You can choose to use ML or to add an extra external ir-eye on the BM1. In both setups the BM1 will respond to the Beo4. But it the latter it will not be possible to use it in a linkroom.
But first I would investigate the alternative naming for the camera input. Go the setup menu, scroll down to connections/sockets. While standing on connnections, press Menu. Then Connections changes to Camera and press Go.