First, change your themes template files (by copying them to a child theme, of course). Remove the H1 tags in the templates, and add them in using your editor.

Second, you can set up extra meta fields to save additional data. If you create one that can save the H1 title that you want to use, you can modify yur themes templates (again, using a child theme) to have them output that value instead of the ormal page title.