< sigh > Welcome to the wonderful world of Pierce Arrow, where you can get ANYTHING customized.
I can think of two explanations why the threads of these to caps are different: If Pierce made the parts in their factory as odd sizes, they made new taps to fit each radiator cap dimension. The more likely explanation is that the radiator and cap were purchased as sets, and Pierce changed vendors at some point between the production dates of these two cars.
Peter, you have a choice; since your dogbone threads match the radiator, it be made usable if you reduce the thread crest (OD of the threads) by about 3/32 of an inch, to match the outside thread of your existing cap. Then make sure the bottom of the threads of the new cap also fits the neck and use it ….. or keep using the cap you have now and let the dogbone find another home, as you mentioned in your first email. Thanks for the interesting problem, it was fun to chew on a while.