Your "Newcomb-like" problem isn't. In the original Newcomb problem there is no situation where both boxes contain a reward, yet the naive CDT makes you act as though there were. In your setup there is such a possibility, so 2-boxing is the strictly better strategy. Any decision theory better make you 2-box.

EDIT: Thanks to those who pointed out my brain fart. Of course both boxes contain a reward in the one boxing case. It just doesn't help you any. I maintain that this is not a Newcomb-like problem, since here 2-boxing is a strictly better strategy. No one would one-box if they can help it.