It turns out that the Kronecker factor doesn't retain a sufficient amount of information about the correlations $\mu(A\cap T^{-n}A\cap T^{-2n}A)$ to prove the desired result.  Indeed, the Kronecker factor is characteristic for the averages, but $\lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \mu(A\cap T^{-n}A\cap T^{-2n}A)$ can be smaller than $\mu(A)^3$ (unfortunately I don't have an example in mind at the moment).  

The large intersection property you ask for was first established by Bergelson, Host, and Kra [1].  Frantzikinakis [2] gave a simplified proof and extended the result to polynomial configurations.  Ackelsberg, Bergelson, and Best [3]
 have extended the result to actions of countable abelian groups, and the introduction there has a nice overview of the history and technical issues involved.

[1] <cite authors="Bergelson, Vitaly; Host, Bernard; Kra, Bryna">_Bergelson, Vitaly; Host, Bernard; Kra, Bryna_, [**Multiple recurrence and nilsequences (with an appendix by Imre Ruzsa)**](http://dx.doi.org/10.1007/s00222-004-0428-6), Invent. Math. 160, No. 2, 261-303 (2005). [ZBL1087.28007](https://zbmath.org/?q=an:1087.28007).</cite>

[2] <cite authors="Frantzikinakis, Nikos">_Frantzikinakis, Nikos_, [**Multiple ergodic averages for three polynomials and applications**](http://dx.doi.org/10.1090/S0002-9947-08-04591-1), Trans. Am. Math. Soc. 360, No. 10, 5435-5475 (2008). [ZBL1158.37006](https://zbmath.org/?q=an:1158.37006).</cite>

[3] <cite authors="Ackelsberg, Ethan; Bergelson, Vitaly; Best, Andrew">_Ackelsberg, Ethan; Bergelson, Vitaly; Best, Andrew_, Multiple recurrence and large intersections for abelian group actions,  [ZBL07471818](https://zbmath.org/?q=an:07471818).</cite>