It is largely speculation of course but it would not surprise me if Quackenbush was involved in providing castings for all these pistols of the 1870-80's period, the Pope, Champion, Eureka, Carey and Cross, as they are interconnected in so many ways.
Certainly the people involved all knew one another. When Bedford moved to Quackenbush's factory the Eureka continued to be made there for a few years, which would be a sensible business decision if the castings were already being made there.
I have never come across a Bedford rifle-air air pistol with cast lettering, as they usually have very small stamped lettering exactly the same size as on the Quackenbush version, and in the same location on the frame. What was the nature of the lettering on the example you have seen?
Unfortunately John Groenwald's book sheds no light on these questions so there is still a lot of research to be done, which I suppose is what makes this field so fascinating for collectors.