It should work. if your wepart produces something like
<iframe src="~/CorporateSite/media/MyMedia/test.html" height="200" width="300"></iframe>. Permission that you set for media library - they are managing permissions. They have nothing to do with the view permissions. I suggest you do a clean test add static html web part put html above with a link to your file and see if it works.
Can you see your HTML file directly in incognito window? If there is not login window- then it has nothing to do with media library - the problem is on the page where you plug your web part.