Not getting an answer online, or in Patrick Hanks’ The Oxford Dictionary of Family Names in Britain and Ireland .
I do get this from Wells, J. C. (2000), Longman Pronunciation Dictionary: Odd pronunciations of proper names – examples: there are four recorded pronunciations of Featherstonhaugh:
/ˈfɛð ərst ən hɔː/ (Featherstonhaw)
/ˈfiːst ən heɪ/ (Feestonhay)
/ˈfɛst ən hɔː/ (Festonhaw)
/ˈfæn ʃɔː/ (Fanshaw)
Let’s us out Feestunhay, which is something else going on, presumably dialectal. We have fɛðərstənhɔː > ˈfɛstənhɔː , eliding the second syllable. So now we need to get from Festonhaw to Fanshaw.
John Gragson’s answer to How did the surname “Featherstonhaugh” get its completely unintuitive pronunciation? is pretty damn ingenious, and I commend him for it. But there’s one factor it misses. It’s just a hint in Hanks’ dictionary, under the entry for Featherstonhaugh: “the surname is often pronounced Fanshaw and may have been confused with Fanshawe.”
Fanshawe is a distinct, equally old surname, deriving from fane ‘a temple or church’ and shaw, ‘a small wood or grove’.
So Festonhaw, which is the recorded shortening of Featherstonhaugh, was somehow garbled further to something like Fesnaw or Feshnaw, as John argues. And then—rather than appeal to a vowel shift (which seems somewhat random) or the phonotactic familiarity of shn vs nsh—we can just say that the garbled Feshnaw sounded so similar to the preexisting surname Fanshawe, that people just conflated the two surnames in speech.
But not in writing. After all, the conflation in pronunciation did not entitle any Fanshawes to any Featherstonhaugh estates.
So the absurd pronunciation of Featherstonhaugh as /ˈfænʃɔː/ has a really simple explanation. They’re not actually trying to say Featherstonhaugh. They’ve switched it to a completely different, easier to pronounce surname, and they aren’t admitting it.