It seems one cannot create a billing user by invitation. It just bugs out. I was able to get around it by creating a normal user, then switching it to billing, at which point I received an error "Billing users may not be in any team" or some such. I'm going to guess this is why one cannot invite a billing user.
In the UI there is "Add to team", which has the only team by default. I suppose if I remove this, it'd work. Anyhow, there is no error text in this UI. It just fails and some elements turn red
Update the flow such that if a user tries to invite a Billing member we should remove the default team
User feedback:
Update the flow such that if a user tries to invite a Billing member we should remove the default team