You have one option only to Embed Image of Dashboard without using server or Public.
A Tableau Creator license would allow you to embed a dashboard hosted on either Tableau Online or Tableau Server (depending on which license you buy) - but users would be required to log in to view it. You would need a Viewer license for each user who would be interacting with the embedded view. If you haven't already, you should contact Tableau Sales, there are licenses that are designed for embedding, and the Tableau Sales team would know more about how those work and what they cost.
Tableau Public could be used as you describe, but would retain the Tableau logo and embedding chrome.