In the Romantic and Victorian eras the spelling "Shakspere", as used in the poet's own signature, became more widely adopted in the belief that this was the most authentic version.
I guess I lean this way. (I do acknowledge what you said about the man himself spelling his own name differently, makes me wonder why...) But I have always been perplexed about different spellings and pronunciations of proper nouns. My name is Dave in every language, I assure you. Sounds like the Elizabethans were all kinds of lax with spelling!