That’s a kind of an odd request, but I reckon you can do it using some HTML elements, like doing a <span class="fakeH1">Fake heading</span>, and then have a CSS snippet simulating the heading formatting.
I’ve not heard of any ways to eliminate the header from appearing in the outline through plain markdown.
For lower level headings you could perhaps get by doing bold and/or italic on a line of its own. Or possibly use HTML like <big>Bigger text</big> or <large>large text</large>, but do remember you can’t put other markdown inside of the HTML elements in any case.
Try using HTML headings (<h2>, etc.) as you suggested. I don’t use Publish, but in local Obsidian they don’t appear in the outline sidebar. Not sure if it works the same in Publish but it’d be easy for you to check.