Unofficially switched PhD supervisor to one at a different institution. Can I get this officially acknowledged?
That is a question that only institution B can answer, but I doubt that they would be agreeable due to your formal link to A. 

What you might be able to achieve is to have supervisor B included in some more or less official way in your process, though, as you say, he/she already declined. 

So, supervisor B doesn't want to be your examiner, but perhaps, subject to supervisor A's sense of things, B could be a co-advisor and also sign your dissertation, giving titles and institution. A may even find it advantageous to have an association with B. 

But that is largely a personality issue. I doubt that either institution would object.