There was no standardized orthography for the Basque language in the SCA's period; all of the recorded Basque names, therefore, were expressed using the pronunciation and orthography of the language that the individual writing the record would have been utilizing, which is generally Latin prior to about 1200, and Spanish (or another Romance language) afterwards.
To provide a parallel situation with which more members of the SCA might be familiar, let us suppose that medieval Irish Gaelic had no written form -- only a spoken form. And suppose that an Englishman were sent to Ireland, and had to write down a person's name in an official record.
And let us say, for this situation, that the person in question is named Giolla Phádraig mac Muircheartaigh, which is pronounced something like gil-PAT-rick muck MUR-tock.
If the written records in England in the era in question are in Latin, the Englishman attempts to Latinize the name. Perhaps he understands that the "muck" syllable in the middle is meant to express "son of." He will then know to use the Latin word "filius" instead, and will to express the other names in as close to a Latinized version as possible, perhaps ending up with something like "Gilpatricius filius Murtocki," so that the grammar fits, even if it isn't quite the same sound any more.
But on the other hand, if the contemporary written records are in English, the Englishman would take his best guess at how the name would have been pronounced, in order to come up with some spelling that would tell the reader how the name would be pronounced through the English system; perhaps the name would end up looking more like "Gilepatrick Macmurtagh."
In either case, if one were to use modern or medieval Irish pronunciation in order to say the name "Gilpatricius filius Murtocki" or "Gilepatrick Macmurtagh," one would not necessarily come up with a name that would sound anything like the original "Giolla Phádraig mac Muircheartaigh”; but if one knows to pronounce the later English version as if it were an English name, it would at least come close to how the medieval name would have been pronounced.
To return to a Basque example, imagine that you are a tax-collector, and that you must record that a man has paid his taxes -- and this man pronounces his name as KAR-duh-luh ih-BAH-nes SSAH-buh-lah.
If your records are primarily in Latin, you will attempt to Latinize the name in some means that will allow you (or some other reader) might be able to pronounce it, so that it can be determined who that person is. The Latinized version would look something like "Cardellus Ibanes Zaballa."
On the other hand, if you are writing after about 1200, in Spanish, you are more likely to attempt to "Romanticize" the name -- that is, to spell it so that it would not look too unusual to a Spanish-speaker, and would, like the Latinized version, give some indication of how the name was pronounced. That version might look more like "Cardiel Yvaynez Çavala.
However, if one goes back and attempts to pronounce either the Latin version or the Spanish version using most dialects of modern Basque, it would sound very little like the original pronunciation. (In fact, a modern Basque version of that name would probably look like "Gardele Ibañez Zabala.")
So, to pronounce the names in this paper, one is more likely to hit upon the proper pronunciation by using the orthography and pronunciation for either Latin or Spanish, rather than to assume that the medieval name would be pronounced precisely as it would have been using the medieval spelling and the modern Basque pronunciation systems.