This appears to be fixed...
I updated Retropie setup to 4.1.10 just now, then did an 'install from source' for lr-pcsx-rearmed.
That fixed it! I can set pad 1 to "analog" and keep the rest as "standard" and this is now correctly detected by any games I play.
I'm super confused by this, because it doesn't look like there have been any commits to this core since I filed the bug. Maybe it was a RetroPie bug?
Can anyone else who's been having this problem please try updating your biz and let me know if it fixes it for you too?