I ended up configuring my CI pipeline to build a Caddy docker image that ships with my website files. The pipeline is also publishing the container image to the Codeberg registry and I apply the new image repo and tag to the Caddy Helm chart I found on ArtifactHub.
The only thing that’s left is to setup the CI to automatically restart the pod when a new image is pushed, so it will always have the latest version.
It was easier than expected and I had a few issues like my stylesheets not being applied and image files not rendering, but it was solved by changing the pathType field on the ingress configuration to Prefix.