Nothing wrong about running http server on your localhost and talk to it. A lot of applications do that. The best thing: you don't need to appease extension appstores, you just ship.

The only nuance is that recent chrome versions treat it as a separate permission, so user need to allow it once.

Yes, native messaging is the "proper" way to do that, but, again, nothing wrong with localhost http server. You have origin headers so you can allow access from your whitelisted website, if necessary.