/'kwɪksət/ is clearly an anglicization of the Spanish spelling.
It's equally clear that such spelling pronunciations have always been very common.
This one, in particular, can be seen in action at the very end of
Canto 13, Stanza 10 of Byron's Don Juan
(a title, incidentally, pronounced /dan'dʒuwən/ by the author, as the poem makes clear)
Redressing injury, revenging wrong,
To aid the damsel and destroy the caitiff;
Opposing singly the united strong,
From foreign yoke to free the helpless native: --
Alas! must noblest views, like an old song,
Be for mere fancy's sport a theme creative,
A jest, a riddle, Fame through thick and thin sought!
And Socrates himself but Wisdom's Quixote?
Note the rhyme: wrong - strong - song alternating with caitiff - native - creative,
and ending in a couplet, with its rhyme boldfaced above. In this couplet,
Quixote has to be pronounced in two syllables, not three,
and the second syllable rhymes with thin sought.
I.e, /'kwɪksɔt/. And from there to /'kwɪksət/ is no distance at all in modern English.