Keep in mind that the responsive menu is available anytime, no matter which header type you choose. You can also trigger it with any button on the page you like. So that might be another solution.
You could display/hide items or elements on specific sites. It is pretty customizable with very little CSS. Optionally some jQuery – if really needed.
From UX perspective, the anchor menu might be a better solution, since it would only require 1 click instead of 2.
