A surviving Muslim Basque kingdom requires a surviving predominantly Muslim Spain.
Unless . . . hmm . . .
Lets say that the Reconquista is somewhat less successful or is being configured differently so there is more Muslim presence up north.
And lets also say that the Crusades are more successful. They take Egypt. The new Egyptian society blends together all its diverse ethnic and social elements into a new Christian religious movement that has significant military fervor. They drive west into N. Africa and then north into Spain. Most ordinary folks and even some elites make their peace with the new conquerors, but some convinced Muslims flee and keep fleeing. They finally retreat all the way to the far north of Spain.
Around this time, however, the *Copt Crusader empire starts experiencing internal dissensions and power struggles, plus the remaining Christian kingdoms in Iberia are jealous of this huge power that is gobbling everything up, so the crusade peters out and Spain dissolves into byzantine power politics.
In the resulting chaos the Muslim Basque area survives, with the Basques going Muslim and the Muslims becoming Basque. Isolated, they form their own distinctive brand of Islam but also see it as part of their ethnic heritage that allows them to stand out from and survive the surrounding Hispanic sea.