Intersecting codes are a classical object in coding theory whose rank-metric analogue has recently been introduced. Although the definition formally parallels the Hamming-metric case, the structure and parameter constraints of rank-metric intersecting codes exhibit substantially different behavior. It was previously shown that a nondegenerate $[n,k,d]_{q^m/q}$ rank-metric intersecting code must satisfy $2k-1 \le n \le 2m-3$, and the tightness of the upper bound was left open. Using the geometric interpretation of rank-metric codes via $q$-systems, we prove that the dual subspace associated with a rank-metric intersecting code must satisfy strong evasiveness properties. This connection allows us to derive new restrictions on the parameters of such codes and to show that the bound $n=2m-3$ can be attained only when $k=3$ and $m\ge 6$. More generally, we show that $n \leq 2m-\lfloor(k+4)/2\rfloor$. Moreover, we obtain a geometric characterization of these extremal codes in terms of scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$. As a consequence, the existence problem for $[2m-3,3,d]_{q^m/q}$ rank-metric intersecting codes is reduced to the existence of scattered subspaces of dimension $m+3$. Using known constructions of maximum scattered subspaces, we derive existence results when $m$ is even. Finally, we prove that $[6,3,3]_{q^5/q}$ rank-metric intersecting codes do not exist for any prime power $q$, thus resolving an open problem posed by Bartoli et al. in 2025.