Analytically (and in Drake), what would the exponential stability requirement look like in this case? Would it be

$$ V(x) \text{ is SOS, } -\dot{V} - \lambda \left( s^2 + c^2 - 1 \right) - \beta V \text{ is SOS } $$

for some small user-selected β > 0, to ensure that the time derivative of V is always proportional to V itself?