How to change input language on iPhone using Mac keyboard in 'iPhone mirror' mode?

I tried it, and while it changes the keyboard language on the Mac, it has no effect here in the mirrored phone.

My inconvenient workaround is to type text on the Mac in the target language, copy, and paste into the phone.

Not OP but I did try and, no, it does not work.

Have you tried the standard shortcut Control Space?