I’m not very familiar with BGP and VRF, however you may have some success using either the routed or p2p NIC types.
These both create a veth pair between the container and the host, and in the case of the routed NIC will add a single-address static route on the host side in the routing table of your choice.
The p2p NIC just creates an unconfigured veth pair between container and host, so you can then do with that what you need.
See
https://linuxcontainers.org/lxd/docs/master/instances#nictype-routed
https://linuxcontainers.org/lxd/docs/master/instances#nictype-p2p