We consider the 2D delta-Bose gas by a smooth mollification of the delta potential, where the coupling constant is in the critical window. The main result proves that for two particles, the approximate semigroups on $\mathscr B_b(\Bbb R^2)$ for the Schrödinger operator with singular interaction at the origin converge pointwise in the initial condition. This convergence extends earlier functional analytic results for the convergence in the $L_2$-norm resolvent sense. The central methods introduced here apply the excursion theory of the 2D Bessel process and the ergodicity of the winding number of planar Brownian motion. The limiting semigroup thus shows both the Kallianpur--Robbins law for additive functionals of planar Brownian motion and Kasahara's second-order law for the fluctuations. As an application, the mode of convergence is extended to the $N$-particle delta-Bose gas for all $N\geq 3$.