Appa-lay-shin isn't an uncommon way to say it there. Appa-latch-in is more common (and correct) but I don't think anyone there would get upset if you pronounced it appa-lay-shin. Half the words they say are mispronounced (Eye-tale-yin for Italian) or made up (sigogglin for askew) or so completely incomprehensible that communication is done more through tone and body language than pronunciation.
There's honestly a good chance they won't know for sure what Appalachian even means: it's a much larger area than they're familiar with. A born and raised West Virginian is probably going to say they're from "the mountains" or "the hills" or "West by God Virginia" or just give you the name of some town with a population of 50 if you include the pet chickens.
Never made this connection before, but that makes sense. WV is mostly Eyetalian-Yankee ancestry: split from Virginia to fight for the North, and there was a lot of Italian immigrants working the mines.
