The Great Vowel Shift is actually not relevant to your question, because it only affected long vowels. There was, however, another vowel change which took place in the late Middle English or early Modern English period -- that is, around the 1400s -- by which the vowel e became a when it was followed in the same syllable by r. This is the sound shift that changed Berney into Barney; it's also the cause of some doublets, like clerk / Clark, university / varsity, person / parson, vermin / varmint.
So the spelling change you've found probably tracks the change in pronunciation quite closely: in that period the sound of your surname was changing to the modern pronunciation. Before the change began, the vowel in the first syllable was probably [e]; standard Modern English doesn't have this exact sound, but the word bear (in standard American pronunciation) might be a rough equivalent to the first syllable of Middle English Berney.