This sounds like a Marked question to me, since that is what you are using to run the actual HTML conversion instead of Scrivener.
There is an MMD command-line argument that will omit header labels: --nolabels. So whichever program you use, that’s going to be what you’re looking to customise the settings with.
To hook that into Scrivener, double-click the “Basic MultiMarkdown” (or whatever you decided to use) Format to edit it, and in the Processing pane, set it up like this:
Field values...
And here are the values for easier copy and paste:
Path:
/Applications/Scrivener.app/Contents/Resources/MultiMarkdown/bin/multimarkdown
Argument:
--nolabels -o <$outputname>.html
Of course, if you have your own copy of MMD installed, you’d want to use that for the path instead of Scrivener’s built-in copy. It’s usually found in:
/usr/local/bin/multimarkdown
With Marked it’s going to be very similar. You’ll find processor settings in the Advanced preference tab.