I grabbed some black Pro's and regular white Eneloop's out and found the same as you as far as size goes. The black Pro's were maybe 10 thousand's of an inch larger in both dimensions and when compared with a standard AA cell, about the same again so not enough to make a difference in the HP-1 at least.
For testing, just running your black Pro's in another working device would probably be a good enough way to test them.
Run them in that device for an hour or so and then pull them and read each cell's voltage fairly quickly. If one is bad, it should be fairly obvious as its voltage will usually be far off from the others.
Another test more like a load test would be if you can get meter probes on each end of each cell while they are powering something like your Sangean or C Crane devices. This may not put a lot of load on the cells but it should be plenty for a simple test while those devices are turned on while you read each cells voltage.
In my HP-1, I can't get at the contacts with a meter probe while the cells are in the radio. The meter probes are so fat that they push the cells away from the radio's contacts. For that matter, when cells are inserted into my HP1, you can barely even see the metal contacts that are a part of the radio as the cells totally block them from view.
If your cells prove good in another device, it has to be a connection problem with the contacts in the HP-2 and the Eneloop Pro's.
DC voltage is DC voltage between the two cell types so something is not making connection knowing the older White Eneloop's do work in your HP-2.
If the cells all prove good, the only other suggestion I would have is maybe take a small pick or pocket screwdriver and bend the Negative (-) battery contact out just a tiny bit further in each of the four cell sockets in your HP-2. This will put more pressure on the cell to push it tighter towards the + contact.
The Positive contact should be a fixed metal plate for each cell so you can't do anything with it other than look for corrosion on all four!
I've also seen these type negative contacts get bent in when cells were inserted in reverse accidentally before. Pulling (bending) them back out usually fixes them so long as you don't bend them out so far that they break!
In the HP-1, each of the four cells has contacts at both ends, so no two cells contact each other directly.
Each of the four cell compartments have the spring steel negative plate in it. I'd think this part of the HP-2 is made the same.
Good Luck!