The Better Business Bureau isn’t actually a government authority or anything. It’s Yelp before there was the internet. So report on their Yelp page, report at the BBB, sure. I’ve seen better traction (ie real public shaming and awareness generating) on local community sites like a Facebook group or NextDoor.
Definitely name names.
My understanding is that most homeless people are ‘local’ to the area. That is, the majority were already residents of a city before becoming homeless and for a variety of common sense reasons would not want to leave. They know the physical and social geography of their area. They have family or other deep ties. They know which shelters are open at which times, which areas to avoid, where to get dinner on a Thursday. Where it’s relatively safe to sleep outside, etc.
Promising a person that if they move they’ll be provided for means they lose whatever social network they have and requires trust that such a promise will be fulfilled and they won’t be stranded in a worse situation.
That understanding is based on a handful of evenings volunteering at a local homeless shelter, conversations with a friend who is heavily involved in the non-profit world of homelessness, and a layman’s interest in housing policy.