Mirroring to iPhone does not respect keyboard

I use a Dvorak keyboard on my Mac. There is no such keyboard on an iPhone, which isn't a problem for me because I'm typing with my thumbs anyway.

The problem is that the new iPhone Mirroring somehow maps the *phone's* virtual keyboard to my *Mac's* physical keyboard, so when I try to type something in an app from my Mac onto my iPhone, the keyboard is QWERTY and not Dvorak.

It would be very annoying after all these years to change the iPhone keyboard to Dvorak to solve the problem, but even that seems to not be an option because the iPhone has no Dvorak layout.

Is there a solution? It would be very handy to not have to break out my iPhone to do iPhone-only tasks, but if I can't type quickly and have to keyboard switch it hardly matters.