Is it possible to disable the header on any page that has a Header 1?
The only way to do this automatically will be to write a macro. This is a bit beyond my macro skills, but this is the process you will need to follow as the only way to switch off a Header or Footer on a single page is by using Sections
You may be able to get some help creating a macro on StackOverflow
You could also follow this process manually.
EDIT: In response to Comment:
Not quite - the "different footer and header" setting is only applicable to "odd and even" or "first page" e.g.:
The only way to mix and match headers is to use section breaks with
I have not tried it to see if it works, but in the "Replace" feature (as in Find and Replace) there is a way to search for a style including Heading 1. You could search by style and Replace with nothing or whatever you like instead.