Well, here we are again with me not being quite able to follow your argument.

1) I agree that (x+y) divides (4y+1)(4y-1)(4x+1)(4x-1).

   In fact (x+y)^2 divides (4y+1)(4y-1)(4x+1)(4x-1)

   But I don't see how you get from there to 

   (x+y) divides (4x-1)(4y-1).  

   Just because (4x+1) and (4y-1) are relatively prime that doesn't mean that (x+y) cannot divide their product.  You might well be right (I have not quickly found a counter-example), but I not understand the logic.

2) Also, we need to have a serious talk about modular division.

    Just because (x+y) divide 16xy + 1, it does not follow that (x+y) = 1 mod 16.  You made that statement, but fortunately you did not (I think) use it further in your argument.  

Take the case of (1,2), which is a solution.

(x+y) divides 16xy+1, but (x+y) = 3 mod 16.