Currently building something similar and following your path. Rendering an iframe and are working on bundling react app into a custom HTML tag and dropping that onto the page. Would be curious to hear more about your experience.
In the end, we built a Wordpress plugin since it turned out that most of our customers used Wordpress. This plugin acted as the reverse proxy. We went a step beyond this and did some cool stuff to let them use a shortcode to render our eCommerce menu within their existing Wordpress template.
One wrinkle with ditching the iFrame was getting our CSS to not conflict with their CSS. I ended up putting our stuff within a shadow DOM, which was a bit of work but ended up working pretty well.