Success! I got the navigation module into the header of my journal style and got it looking the way I wanted on mobile too. Just as I was doing it, Dreamwidth did a code push that standardized the availability of the option to put navigation in headers for the remaining styles.
I'm still using Practicality - or rather, I'm using a hacked-and-modded fork of Practicality, because there are some overrides you can only do with a layout layer - and Practicality already contained this option, although I wasn't using it before and its default appearance wasn't what I wanted anyway. Even though I knew what I wanted to do more or less, I went through all the Dreamwidth styles and checked their appearance on mobile for ideas first (hopeful of getting a nice bit of code I could borrow, but no such luck), so I know that all of the ones which do let you put more stuff in the header condense a bit awkwardly and take unnecessary space on mobile; that's why I had all that stuff out of the header before, because it was the only easy way to make the header less unwieldy. The control strip at the top already takes a ton of space on the screen. (Trying to style that smaller is my next goal. I've seen custom user-created styles which put the control strip on autohide, behind a button, but the button didn't work both ways on mobile, ie you couldn't tap to hide it again.)
I'm still using Practicality - or rather, I'm using a hacked-and-modded fork of Practicality, because there are some overrides you can only do with a layout layer - and Practicality already contained this option, although I wasn't using it before and its default appearance wasn't what I wanted anyway. Even though I knew what I wanted to do more or less, I went through all the Dreamwidth styles and checked their appearance on mobile for ideas first (hopeful of getting a nice bit of code I could borrow, but no such luck), so I know that all of the ones which do let you put more stuff in the header condense a bit awkwardly and take unnecessary space on mobile; that's why I had all that stuff out of the header before, because it was the only easy way to make the header less unwieldy. The control strip at the top already takes a ton of space on the screen. (Trying to style that smaller is my next goal. I've seen custom user-created styles which put the control strip on autohide, behind a button, but the button didn't work both ways on mobile, ie you couldn't tap to hide it again.)