In this paper we study the potential theory of Dirichlet forms on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-α}\mathcal{B}(x,y)$ and the killing potential $κx_d^{-α}$, where $α\in (0, 2)$ and $\mathcal{B}(x,y)$ can blow up to infinity at the boundary. The jump kernel and the killing potential depend on several parameters. For all admissible values of the parameters involved and all $d \ge 1$, we prove that the boundary Harnack principle holds, and establish sharp two-sided estimates on the Green functions of these processes.