I think it is possible. But you have to use a lot of code to update the custom field value to each WordPress setting after saving the settings page. I do not see the benefit when doing that. Instead, you can create your own settings page with custom fields and get that field value when you build the page/post on your site.
Thank you for your answer. I feel you are correct. I have been able to cheat the system and edit some fields when I give my custom field the ID of an existing settings field, but this does not work consistently.
I will just create my own settings and find ways to incorporate that into my theme. Thank you:-)