That makes a lot of sense, thanks for explaining for use case. As mentioned, this is not easily possible at the moment, but as it's a very useful feature, feel free to create a feature request in the Kibana Github repository: https://github.com/elastic/kibana/issues/new?template=Feature_request.md
You mentioned you developed a Kibana visualization - does this mean you wrote your own Kibana visualization plugin? If that's the case you could use a similar custom plugin to apply some CSS to the page to hide the three dots if the page is rendered in an iframe. This is not a supported solution, but I worry it's the best solution to get it running right now.
Also, I haven't done so already, maybe give Kibana Canvas a look - it gives you more control over the visual aspects, but it depends on your specific case whether it's a viable replacement.