Sounds like using OVN (https://www.ovn.org) from Openvswitch which LXD has recently added support for would be appropriate here.
It allows multiple virtual networks to be setup (across multiple machines) and manages the tunneling of traffic between machines (using geneve tunnels). It then uses OVS on each host to connect to the instances via veth pairs (like a normal native bridge does).
A simple single-node setup can be used to experiment with pretty quickly, we have a basic guide as part of the ovn network type docs here How to set up OVN with LXD - LXD documentation
Each network is given a virtual router, and this provides NAT, DNS relay to specified DNS servers, internal DNS, DHCP v4/v6 and IPv6 router advertisements.
With LXD ovn networks we have a concept of a uplink network using the network property, which is then used to provide external network access for the virtual router (with or without NAT), and we support using the existing lxdbr0 network as a parent network for single-node OVN networks (albeit it with a small modification to assign certain IPs from lxdbr0 for OVN use).
For larger deployments we also support assigning a physical NIC for OVN parent network usage using the physical network type Physical network - LXD documentation
OVN uses two specialised database services that all the compute nodes connect to. These are called the “northbound” database (which LXD uses to define the logical network config) and the “southbound” database (which OVN populates from the northbound database) to provide the openvswitch flow config for each compute node.
These databases can be run on a separate node/instance than the compute nodes, and can be independently clustered for HA.
Assuming a single database node, with multiple compute nodes a basic setup would be like this:
On the database node:
# Install central database services.
sudo apt install ovn-central -y
# Expose Northbound and Southbound databases onto network (note: no security).
ovn-nbctl set-connection ptcp:6643:0.0.0.0
ovn-sbctl set-connection ptcp:6642:0.0.0.0
On each compute node:
# Install OVN compute node services.
sudo apt install ovn-host -y
# Connect openvswitch to OVN southbound database.
sudo ovs-vsctl set open_vswitch . \
external_ids:ovn-remote="tcp:<database node IP>:6642" \
external_ids:ovn-encap-type=geneve \
external_ids:ovn-encap-ip=<compute node local IP used for tunneling>
# Inform LXD where the northbound database is.
lxc config set network.ovn.northbound_connection=tcp:<database node IP>:6643
# Check the `br-int` OVN integration bridge exists.
sudo ovs-vsctl show
Then you can configure LXD OVN networks.