@Lisa7 that's a great idea. After saturating my social network with this question this afternoon I did eventually figure out how to edit the "Permit / Tax ID" field (for me it was in the "Location" section, same page as the map but below the fold and out of sight), so I think I have myself covered. It sounds like that "Permit / Tax ID" field isn't in the same place when @Monica4 edits her host listing, maybe due to regional differences (I guess information in the "Location" section would vary from place to place - though you'd think a section called "Location" would deal with, ya know, LOCATION - not "Location + Legal compliance").
But a screen shot with the id displayed seems like a great workaround for folks who can't find a data field to enter their legal registration info into.
Obviously Airbnb is pretty keen to prevent the direct exchange of phone numbers, website URLs, email addresses and such via host listings, so I'm sure IN GENERAL I'm sure they do what they can to prevent numbers/addresses/etc from being posted in photos... but I suspect they wouldn't aggressively remove photos of *compliance codes*. But who knows, maybe they have bots that indiscriminately remove photos that display numbers. Would be worth a shot if there were no other option.
Thousands of Airbnb hosts must be dealing with this issue in cities all over the world. I'm really surprised there isn't a more straight-forward, generic workflow that guides hosts thru legal compliance and data entry of required registration info. Maybe the SF STR registration codes are especially tricky because they are 7 digits and resemble local phone numbers?
All in all, way too much user effort to do something basic (and legally mandatory). But hey, no ever said Airbnb was the iPhone when it came to intuitive usability.