there is an equal chance that Check B is half or double Check A.

No, there is not. Cheque B is $8. It has no chance of being $2.

On the other hand, my best guess, given the limited information that I have, is that there is an equal chance that cheque B is half or double cheque A. These odds are a reflection of my uncertainty about the situation; they do not magically invoke the possibility that cheque B might actually be $2. It isn't, it's $8. The outcome, when I finally settle on one of the cheques, is not determined by my ignorance but by the actual facts. If I switch, I will get $8 not $2.

And why, by the way, would I wish to employ a smoke screen to add further confusion?. It's fairly standard in this sort of puzzle. Part of the skill in solving them is to identify the relevant facts.