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The non-interacting and non-Hermitian, parity-time ($\mathcal{PT}$)-symmetric Anderson model exhibits an exceptional point (EP) at a non-Hermitian coupling $g=1$, which remains unrenormalized in the presence of interactions (Lourenco et al, arXiv:1806.03116), where the EP was shown to coincide with the quantum critical point (QCP) for Kondo destruction. In this work, we consider a quantum dot hybridizing with metallic leads having Rashba spin-orbit coupling ($λ$). We show that for a non-Hermitian hybridization, $λ$ can renormalize the exceptional point even in the non-interacting case, stabilizing $\mathcal{PT}$-symmetry beyond $g=1$. Through exact diagonalization of a zero-bandwidth, three-site model, we show that the quantum critical point and the exceptional point bifurcate, with the critical point for Kondo destruction at $g_c=1$, and the exceptional coupling being $g_{\scriptscriptstyle{EP}} > 1$ for all $U\neq 0$ and $λ\geq 0; λ\neq U/2$. On the line $λ=U/2$, the critical point and the EP again coincide at $g_c=g_{\scriptscriptstyle{EP}}=1$. The full model with finite bandwidth leads is investigated through the slave-boson approach, using which we show that, in the strong coupling regime, $λ$ and interactions co-operate in strongly reducing the critical point associated with Kondo destruction, below the $λ=0$ value.