Customized header with functional mobile menu?

This should be easy, and able to be done in Wix.
But as with many things, it’s not. So, wondering if it’s possible to force something with code.

I want a customized look in the header on my home page (an image), but then standardize/simplify the header on the subpages (simple color).

But: Can’t have multiple header designs/styles in Wix, so the only option is to “Turn off headers/footers” on the home page, then manually simulate a header/footer with Strips.

I’ve done that, and Copied/Pasted in the header elements manually into the strip - the logo and the menu element. Looks great and works - EXCEPT…

The menu element only works on the web view. It doesn’t transfer to the mobile view, and forces it way back into the “Header,” (hidden) - EVEN THOUGH it’s literally manually pasted in the Strip that is part of the Page items.

Thus: NO menu on the mobile home page. No hamburger. Nothing. Not accessible.

(So, Wix has forced my hand here. I literally can’t make the design I want, because they’ve made their “work-around options” incompatible with each other.)

So:
Customized header look on the home page,
with a menu that works across all pages - web view and mobile view.
And sharks with frickin’ laser beams.
Can this happen?

How about setting up another home page for mobile only.
https://support.wix.com/en/article/corvid-tutorial-redirecting-mobile-visitors

You can set up pages that have elements etc appear only in mobile versions by using the Wix Window API and the formFactor function as shown here.
https://www.wix.com/corvid/reference/wix-window.html#formFactor
https://support.wix.com/en/article/corvid-tutorial-displaying-elements-in-mobile-only
https://support.wix.com/en/article/corvid-writing-code-that-only-runs-on-mobile-devices

You can vote for that option here.
https://support.wix.com/en/article/request-adding-two-different-menus-to-your-site-1395713

Plus, as that page states, have a look at anchor menus too and see if they are visible for you too,
https://support.wix.com/en/article/anchor-menu