Without loss of generality, $x=0$. 
Since $T(f_1,\dots,f_n)$ is $n$-linear in $(f_1,\dots,f_n)$, you can express each $f_i$ as a mixture of harmonics $\exp(it\cdot)$, $t\in\mathbb R$ (with, in general, complex coefficients), so that $T(f_1,\dots,f_n)$ is expressed as a mixture of the values $T(\exp(it_1\cdot),\dots,\exp(it_n\cdot))$ with real $t_1,\dots,t_n$. 
In turn, to express $T(\exp(it_1\cdot),\dots,\exp(it_n\cdot))$, you can expand the product $\prod_{j=1}^n(e^{it_jx}-e^{it_jy})=\prod_{j=1}^n(1-e^{it_jy})$ in the numerator of the integrand in $T(\exp(it_1\cdot),\dots,\exp(it_n\cdot))$ and then proceed as in Appendix A in https://arxiv.org/abs/1603.07365, based on Lemma A.1 there (cf. Proposition A.4 there).