Hi ,
As of now , You cannot use HTML to make changes to your site using In-Browser Editing and also there is no option available at present to lock the changes you have made and make them permanent forever.
The only workaround to make your changes permanent is to open your site in Muse. Make the changes and from the site settings option un-check the in-browser editing option. And then publish your site again. However once you do this, you wont be able to use the In-browser editing option in the future with your site.
However, I request you to post your ideas regarding this to Ideas for features in Adobe Muse
Thanks,
Rohit Nair