I used the Wayback Machine to find an archived copy of the BuddyPress forum. This is what I had posted on there:
Well, I found a workaround. For me it wasn’t a problem with the toggle … kind of.
I found that in the function invite_anyone_bypass_registration_lock(), it was this check that was causing the failure:
$bp->current_component != BP_REGISTER_SLUG
This seems to be because (i) I had changed the slug of my BP registration page and (ii) my registration page is a child page of another page.
Workaround: I set my slug to be “register” and set the registration page to be a main page — and lo-and-behold, email invitations work again even with site registration disabled.
I don’t know if that’s a bug in WordPress, BuddyPress, or this plugin. Hopefully this workaround will help others to have a functional site.