So this has been mildly annoying for long enough I'm finally writing it up, although I will immediately point out that nothing in MRBP is actually blocked or unavailable because of this oddity. It only results in some extra clicking required, and it looks weird.

What Works:

On my main character -- more precisely, the first character on my account to enter The War Within content, first to unlock each faction, first to complete the storyline, etc -- everything via the minimap is displayed correctly. The minimap icon itself is updated to be the TWW icon, left-clicking it displays TWW faction summary, and so on. Here's what it looks like when the mouse is hovering over the minimap icon:

(The only unusual bit is the fact that TWW has a gray background, but I'm guessing that's just visual indicator for "this is the xpac you're currently in".)

The Weirdness:

Every other character in TWW is stuck displaying Dragonflight as the "most recent unlocked" expansion. Here's what hovering the minimap icon looks like for a level 80 alt, sitting in Dornogal, having completed the campaign storyline several weeks ago:

The icon is the Dragonflight icon, the labels are all for DF, left-clicking it open the DF factions, etc. Note that the line for TWW still has the gray background, so something is correctly deciding that it's in TWW content.

Hovering over individual lines on that flyout menu to display xpac-specific summaries also works. And for the characters who haven't unlocked a given xpac content, it correctly shows the "go complete Some Quest Name to see this stuff" hint in red text. For example, the alts who never went to Shadowlands get only the warning that they need to go choose a covenant. But no such line appears for these alts; here's the summary displayed for the same alt as previous:

So all the information is still there, but it's never detecting TWW as "the current content" expect for the main toon. All other toons get DF by default. And I can click around to get to the correct TWW, but I'd like to figure out what's causing this minor annoyance and do whatever I can to fix it before more alt+expansion combinations go wonky. :-)