You can configure Excalidraw to automatically export your drawings as PNGs or SVGs in the same folder as your drawing, and keep them up-to-date as your drawing changes. You can then embed the PNG in your page instead of the Excalidraw drawing, and have an up-to-date image that (I assume) Publish will handle gracefully.
Caveat: I don’t use Publish, but I’ve used this method with other plugins like Advanced Slides and it works well for me.