The Pismo is certainly known to recharge backup cells by removal of the main battery. I don't know about the Wallstreet, but you could give it a try and post the results for the record. If, after a day or so of being plugged in without the main battery, it sleeps and blinks for even a little while when mains power is removed, you have a success on your hands. It might even be good to 'cycle' those cells by letting them fully discharge — and then repeating the process. That's what I did with my Pismo, and it seemed to make the machine very happy.
I used a Wallstreet for years, but never tried this tactic on it. However, what I can tell you is that I have found that Wallstreet backup batteries really do die, and that they will not be rehabilitated whatever you do with and for them by way of spending lots of loot on the acquisition of fresh main batteries. 1xx series backup cells, by contrast, seem to have the DNA of the Energizer Bunny.... They don't make them like they used to?