No, this is is hopeless, as becomes clear when you write $$ \textrm{Cov}(X,Z)+\textrm{Cov}(Y,Z)=E(X+Y)Z . \tag{1} $$ Your assumptions don't restrict $U=X+Y$ much, you only know that $EU=0$, $\textrm{Var}(U)=2$ and $U$ is 2-divisible.

More to the point perhaps, you can just take $Z=(X+Y)/\sqrt{2}$ to obtain a counterexample, then (1) equals $\sqrt{2}$. This is also the largest possible value, by Cauchy-Schwarz.

No, this is is hopeless, as becomes clear when you write $$ \operatorname{Cov}(X,Z)+\operatorname{Cov}(Y,Z)=E(X+Y)Z . \tag{1}\label{463874_1} $$ Your assumptions don't restrict $U=X+Y$ much, you only know that $EU=0$, $\operatorname{Var}(U)=2$ and $U$ is 2-divisible.

More to the point perhaps, you can just take $Z=(X+Y)/\sqrt{2}$ to obtain a counterexample, then \eqref{463874_1} equals $\sqrt{2}$. This is also the largest possible value, by Cauchy–Schwarz.