If I had of known they were flat I wouldn't have ordered them. There is something different though with this button cover because these batteries work in both my other REOs with metal button covers. It's definitely the button cover that is the issue.
On 13 of my metal Reos, 5 have black delrin, 1 has a white delrin, 2 have brass and 5 have aluminum Reo button covers. None of them are an issue at all when firing. But then I also only run button top batts in my metal Reos.
I also have a bag with a few new spare buttons, so took some measurements for you. The recess in all the buttons is the same, metal or delrin, .100". With that info if you have a vernier caliper or depth gauge you can check that black delrin button to see if it is factory spec. Or compare it to another with a toothpick, etc as a depth gauge. If it is the right depth the button is not at fault.
Might you have a partially collapsed spring in that Reo? Using flat top batteries if the spring is not holding the batt up high enough for the positive contact to get a good connection might be why in only that Reo. Another possible flat top batt thing, check for a cracked positive contact from mashing the button.
Ed... my experience with button installation is the brass are the tightest so take more force to install, the delrin and aluminum press and lock on very easily.