I don’t think it’s plausible by any stretch of the imagination that there were only 350 (!!) surviving Jews in all of (western?) Europe in 1351, and yet there were another 999,650 Jews living elsewhere (North Africa and the Middle East? Where else were there any significant number of Jews in 1351?).
Iberian Peninsula had the most Jews at the time I believe, and there were some significant populations in the Near East (IIRC). These were not Ashkenazi though.
In and case, that 350 number is totally absurd.