$$\int \partial_1 (\langle x\rangle^{\delta-1} x_1 u^2) \mathrm{d}x = 0 $$ So $$ \int [ (\delta - 1) \langle {x}\rangle^{\delta - 3} |x_1|^2 + \langle{x}\rangle^{\delta - 1}] u^2 ~\mathrm{d}x = -\int 2 \langle x\rangle^{\delta - 1} x_1 u \partial_1 u ~\mathrm{d}x $$ so by C-S $$ | \mathrm{LHS} | \leq 2 \left( \langle x\rangle^{\delta - 3}|x_1|^2 u^2 ~\mathrm{d}x\right)^\frac12 \left( \langle x\rangle^{\delta + 1} (\partial_1 u)^2 ~\mathrm{d}x\right)^\frac12 $$
So as long as $\delta > 1$ your desired inequality holds by one further application of Young's inequality to the RHS and absorbing the undifferentiated factor on the left.
When $\delta \in (-1,1]$ your desired inequality clearly does not hold since the RHS is non-increasing when translating a function of compact support far away toward infinity, while the LHS increases by the positive weight.