live-boot initramfs hooks check the /live/filesystem.squashfs file by default. And you can add files to initramfs simply by appending it to the existing file (no need to unpack & repack).
I already do this for quite some time and this is roughly the code I use:
echo live/filesystem.squashfs | cpio -o -H newc | gzip -1 > $live_initrd
cat $org_initrd $live_initrd > $new_initrd
Replace the original initramfs image with the newly created one and there is no need for any other changes, it'll just work.