Maybe some sort of back door entry might prove fruitful.
West Riding, Yorkshire was loaded with hundreds, perhaps thousands of Beaumonts, and one Henry Beaumont of Crosslands fits the panorama rather well timewise. His will was probated and is/was a public document. He left half his estate to 3 of his illegitimate children; he surely gave them some kind of names, and it could well have been Fayerbanke.
A review of Henry Beaumonts will might provide some insite.
Good luck all.