Can having too small of a roll or yaw stiffness destabilize the dutch roll mode?
To estimate whether the dutch-roll mode goes unstable, we can assess the real-part of the dutch-roll eigenvalue: negative is stable and positive is unstable. From Etkins, Dynamics of Flight, one illustrative estimate (although not necessarily numerically accurate) is as follows:

$$n_{DR}\approx\frac{1}{2} \left[ \frac{Y_v}{m} + \frac{N_r}{I'_z} + \frac{I'_zL_v}{I'_xN_v}\left(\frac{N_p}{I'_z} - \frac{g}{u_0} \right) \right] $$

where $n_{DR}$ is the real-part of the eigenvalue, $Y_v$ is the dimensional side force sideslip derivative, $N_r$ is the dimensional yaw moment damping, $L_v$ is the dimensional negative roll stiffness (dimensionless form: $C_{l_{\beta}}$), $N_v$ is the dimensional yaw stiffness (dimensionless form: $C_{n_{\beta}}$), $N_p$ is the yaw-roll cross-coupling moment, $u_0$ is the trim airspeed, $g$ is gravitational acceleration, $m$ is aircraft mass, all the $I'$ are inertia terms about the respective axes. Note that all the inertia terms are about the principle axes for simplicity.

The first term ($Y_v$) is always negative, and so is the second term ($N_r$). Ignoring $N_p$ for now, positive roll stiffness requires $L_v$ to be negative, and positive yaw stiffness requires $Y_v$ to be positive; this means that the third term is net positive. Therefore, we can draw the following conclusions:


Dutch-roll damping worsens at lower airspeed
Roll stiffness worsens the dutch-roll damping
Yaw stiffness improves the dutch-roll damping


To answer your question: yes, it's entirely possible for dutch-roll to be unstable due to sufficiently low yaw stiffness.