Let $G_1$ denote the incidence graph of the complete graph $K_{q+1}$. We study limited augmented Zarankiewicz numbers in this family by combining exact 0--1 ILP computations for the smallest cases with a constructive search procedure followed by exact admissibility verification in the larger cases considered here. We obtain $$z_L(6,4)=14,\qquad z_L(10,5)=26,\qquad z_L(15,6)\ge 43,\qquad z_L(21,7)\ge 64,\qquad z_L(28,8)\ge 88.$$ The first two values are exact. The three lower bounds arise from explicitly verified admissible families with $|E_2|=13$, $|E_2|=22$, and $|E_2|=32$, respectively; the families used to obtain these bounds are nondegenerate in the sense of [8]. In each case, the resulting value improves the corresponding classical Zarankiewicz number and hence strengthens the available lower bounds for BSR(m,n) within this family.