Mohammed comes in at 571. Martel in 688. The formal Moorish invasion comes as late as 714 or thereabouts. However, Islamic missionaries had already been there, and in SW France. Islam was not spread entirely by the sword.
It is highly doubtful that the Moslems in France much appreciated Martel and his gang nosing around.
The levels of civilization of the two parties were quite pronounced at the time, and any modern person would have much preferred living in the Moslem areas to living in almost anything else in France.
San Cho Noe came from Cornwall. Although many other tribes lived in the rump Visigoth state in Galicia (that is, Carvajal), the number of folks moving in from Brittain probaly exceeded the existing population substantially.
If you want to call the Salicians "Franks", go ahead. They certainly didn't have a Frankish empire - and rest assured Louis XIV relished having Leinsterian ancestors much more than he did "Frankish" ancestors.
Not all Islam was spread by the sword...ok fine no it was not but that has nothing to do with the point I make.
You state that the Muslims in Southern France did not care to have Martel "nosing around" that is disingenuous. It intimates that the Mulsims in Southern France were welcome in the first place when in fact they did come as invaders a few missionaries does not negate the fact that the muslims were invading Southern France. Or I suppose that the battle at Poitiers was just Martel slaughtering a bunch of helpless innocent unarmed Muslims who happened to march their looking for a home. As for what to call the french at the time I suppose one could call them the Carolingian Dynasty Which was really started by Martel but even Martels father Pepin was a tremendous power in France before Martels defeat of the muslims.
Nevertheless the Merovingians were followed by the Carolingians and they by the Capetians but all were considers Franks.