Variance of random variable decreasing in parameter

Your probability measure is a product measure, so by $$text{Var}_y(langle z,Xrangle) = sum_{i=1}^nz_i^2text{Var}_{y_i}(X_i)$$ everything reduces to the 1d case. Let $q_y(dx)=e^{xy-varphi(x)-C(y)}dx$ be one of the marginals, where $yinmathbb{R}$ and $C(y)$ is chosen such that $q_y$ is normalized, and denote by $U_y$ the 1d r.v. with distributon $q_y$. It can be shown that $C(y)-C(0)$ is the cumulant-generating function of $U_0$, and $text{Var}_y(U_y)=C''(y)$. Thus the variance has a local maximum at $y=0$ iff the third cumulant $kappa_0^{(3)}$ of $U_0$ vanishes and the fourth cumulant $kappa_0^{(4)}$ is negative. Indeed, if we denote by $m_y^{(k)}$ the $k$'th moment of $U_y$, we have $$kappa_0^{(3)}=m_0^{(3)}-3m_0^{(1)}m_0^{(2)}+2left(m_0^{(1)}right)^3=0,$$ because $p_0$ is symmetric and therefore the first and third moments are zero. Similarly, dropping the odd-numbered moments, we have $$kappa_0^{(4)}=m_0^{(4)}-3left(m_0^{(2)}right)^2.$$ Therefore assuming that $varphi(x)>lambda x^2$ a.e. for some $lambda>0$ $(star)$, then begin{align*} kappa_0^{(4)}&=int_{mathbb{R}^2}left(x^4-3x^2y^2right)e^{-varphi(x)-varphi(y)-2C(0)}dx dy \ &< e^{-2C(0)}int_{mathbb{R}^2}left(x^4-3x^2y^2right)e^{-lambda x^2-lambda y^2}dx dy\ &=0, end{align*} because the last expression is proportional to the fourth cumulant of a Gaussian. In order to show that this is the global maximum, let's show that $C''(y)$ is concave, i.e. $C^{(4)}(y)<0$ for all $yinmathbb{R}$. In fact, since $$C(y)=logint_{-infty}^{infty}e^{xy-varphi(x)}dx,$$ $C^{(4)}(y)$ is the 4'th cumulant of $U_y$, i.e. begin{align*} C^{(4)}(y)&=m_y^{(4)}-4m_y^{(3)}m_y^{(1)}-3left(m_y^{(2)}right)^2+12m_y^{(2)}left(m_y^{(1)}right)^2-6left(m_y^{(1)}right)^4\ &proptoint_{mathbb{R}^4}left(x_1^4-4x_1^3x_2-3x_1^2x_2^2+12x_1^2x_2x_3-6x_1x_2x_3x_4right)e^{sum_{i=1}^4(x_iy-varphi(x_i))}dx\ &<int_{mathbb{R}^4}left(x_1^4-4x_1^3x_2-3x_1^2x_2^2+12x_1^2x_2x_3-6x_1x_2x_3x_4right)e^{sum_{i=1}^4(x_iy-lambda x_i^2)}dx\ &=0, end{align*} again because the last expression is proportional to the fourth cumulant of a Gaussian (with non-zero mean).

$(star)$ This is slightly different than your 'increasing convexity' assumption, but seems sufficiently close.