It depends on the storage backend being used for the LXD container.
overlay2 is incompatible with ZFS and there’s no sign of that improving any time soon.
But if your container is stored on just about anything else, then overlay2 should work fine, especially since you’re dealing with privileged in your case.
For unprivileged, it will work too, at least so long as you run on a kernel that allows unprivileged overlayfs (Ubuntu does, not sure what other distros do these days).
Still in the unprivileged case, you’d also want to set security.syscalls.intercept.mknod=true and security.syscalls.intercept.setxattr=true to avoid some issues unpacking some Docker layers.