If the Remington caps needed to be pinched to stay on then they clearly are not too tight and if the CCI caps worked fine with the old spring but fail with the new then clearly the spring is the issue, not caps or nipples. C&B revolvers do require a heavier hammer fall than do centerfire revolvers because the hammer impact is spread over a much larger area, the whole head of the cap, as opposed to just the tip of a firing pin.
I'm not familiar with the Whitney revolver, don't know if it has a tension screw like the Remington but you do need to find a way to put more tension on the spring, perhaps a shim at the lower front of the spring to angle it back more?