Least Square Fit - Transfer Function
As @Chu didn't want to answer the question as is possible to see on question comments, I felt free to answer this to give it a finish.



The problem with the fit, is that the inverse Z transform was wrong. That's the right transformation:

$$y(k) = \frac{a_0}{b_3}u(k+2) + \frac{a_1}{b_3}u(k+1) + \frac{a_2}{b_3}u(k) - \frac{b_0}{b_3}y(k+3) - \frac{b_1}{b_3}y(k+2) - \frac{b_2}{b_3}y(k+1)$$

Result: