After responding to about ten requests thus far (with about another twenty to go at this time), please be sure to let me know which proof you want. [Suggestion: Request the first CLT one first. If you could follow it, then request the second, and so on.]
Also, for the most part, this is for the Teacher Only. Furthermore, those who are new APStat teachers should focus primarily on the course, with the authors of all three of the three most-used text- books out there in the APStat Forum (Dave B, Chris O, and Daren S) willing to answer any questions you have concerning The Course and The Course's Subject Matter (including, of course, how to introduce the CLT).
Thus, in a nutshell, it's the APStat Course First! [Those who are veteran APStat teachers should be the primary "requesters" of these proofs at this time.]
-- David Bee
PS: When I taught APStat in its first year, even though I had an MS in Applied Stat, I still spent the entire summer of 1996 giving the textbook a cover-to-cover reading.