I want to have an HTML form embedded in another form like so:
What you have described will not work.
One workaround would be to create two forms that are not nested. You would use hidden inputs for your original parent form that duplicate the inputs from your original nested form. Then use Javascript/DOM manipulation to hook the submit event on your "parent" form, copying the values from the "nested" form into the hidden inputs in the "parent" form before allowing the form to submit.
Your form structure would look something like this (ignoring layout HTML):