A local strict subharmonic peak function globalizes

Choose $\eta >0$ with $q\le -\eta$ on $\mathrm{\Omega }\cap \mathrm{\partial }W$, and then choose a constant $c>0$ so large that $cq\le -1$ on $\mathrm{\Omega }\cap \mathrm{\partial }W$. The function $cq$ is still subharmonic on $\mathrm{\Omega }\cap W$ by [L1], remains negative there, and still tends to $0$ at $\zeta$.

Define [L1, L2, step 1.1] $b(z)=\{\begin{array}{ll}\max \{cq(z),-1\},& z\in \mathrm{\Omega }\cap W,\\ -1,& z\in \mathrm{\Omega }\setminus W\mathrm{.}\end{array}$ Inside $\mathrm{\Omega }\cap W$ the function is subharmonic by [L1]. On the seam $\mathrm{\partial }W$ one has $cq\le -1$, so the inside limsup is at most the outside value $-1$; [L2] therefore glues the inside and outside pieces into a global subharmonic function on $\mathrm{\Omega }$.

The function $b$ is negative on $\mathrm{\Omega }$, tends to $0$ at $\zeta$ because near $\zeta$ the maximum chooses the $cq$ branch, and is identically $-1$ outside $W$, so it stays uniformly below a negative constant away from $\zeta$. Thus $b$ is a global barrier at $\zeta$.