Nothing is impossible and Incus is flexible enough to get you there. It requires of course some special configurations like in your case. Having more than one interface attached to an COI container requires a manual approach but it is now much simpler to archive. (Add second network interface to OCI instance) was written around version 6.4 and a few things have changed since than, take a look at Incus proxy bind to host address issue how this can be accomplished today.
It still requires to write a script which performs the required steps but it doesn’t requires to touch the OCI image anymore. It mainly depends how the OCI starts and what kind of hooks are available to mount your script into it. You just have to make sure it runs before the OCI application is started.
From a quick look at official caddy OCI it is Alpine based and starts caddy directly. In this case a script is needed to configure the second NIC first and start caddy. Shouldn’t be that difficult…