This problem sounded vaguely familiar, and I just found out why.
This is logged on the (old) CiviCRM issue tracker as CRM-17451. The answer was that someone would have to write this functionality or work with the core team to do so. However, I also see that I found a workaround back in 2016. From the page:
In non-technical terms: Profiles secretly have a contact type they're
associated with. You can see this if you try to add an
organization-specific field to a profile, then an individual-specific
field. You'll get the error "Cannot add or update profile field type
Individual with combination of Household or Organization". That's
because Civi has decided this is a profile associated with
Organizations. However, when you don't have any contact type-specific
fields on a profile, your profile defaults to a contact type of
"Individual" for many purposes. So individual contact subtypes are
The workaround: Add an organization-specific field (like "Organization
Name") to your profile, and this problem goes away.