So the Basques were only converted to Catholicism in around 1600, but Europeans have had contact with the new world for a century by that time - with Basques playing a major role in contact with the Americas. Was there any evidence of Basque paganism being practiced in the Americas, or possibly Basque pagan beliefs being passed on to or adopted by Native Americans?
Sorry, what is your source for saying the Basque only converted to Christianity in such a late date? There are plenty of medieval churches in the Basque Country, documents accrediting the lords to be catholic (all sorts of religious incipits or explicits), etc. The whole notion of the Basques converting in the dates you suggest is rather bizarre.
Testaments and notarial documents would also contain religious references, indications for the number of masses to be held in honour of the deceased person, what church would they like to be buried in, donations to churches or monasteries, etc.
It is true that the first book in Basque is from the 16th century (by father Beñat Etxepare) and that it is religious in nature, but that does not mean Basques were pagans. It is simply a matter of adequately teaching religious doctrine for the people not familiarised with Spanish language, or without a knowledge of Latin (the vast majority of the populace).