Can I return Chanel to any Bloomingdales?

It depends on the store.
Of course there are only two stores that have a Chanel boutique.

But when it comes to other stores, it depends on the store. You can call the store and ask to speak with the designer handbags manager and ask them. Some stores don't want a Chanel return because they can't identify/authenticate it well enough; other stores don't seem to mind.
 
I purchased a Chanel bag from Bloomies over the phone, West Coast and was able to return it to a NJ Bloomingdales, but I called in advance to make sure it was allowed. hth