Something I just thought of an hour ago:
For simple cases of instruments sharing staves (like Sopranos and Altos together) you can duplicate the Instrument Names font and edit the duplicate in FontForge, copying the “S” and “A” glyphs into the “" slot (or another character that you're incredibly unlikely to ever use in an Instrument Name). You'll need to nudge both glyphs up and down so they're appropriately centred on the baseline. Then save the duplicate font (making sure to rename it in FontForge) and set your Instrument Name paragraph font in Dorico to the new duplicate font. Then set your instrument/player name to whatever wacky character you've used ("” for example).
I’ve cobbled up the example, but the point is it works.
Disclaimers: there’s loads of FontForge documentation on the net so don’t ask for FontForge help here please! Expect it to mess up things like the Player/Instrument list wildcard - it inevitably will.