We consider optimal sensor placement for hyper-parameterized linear Bayesian inverse problems, where the hyper-parameter characterizes nonlinear flexibilities in the forward model, and is considered for a range of possible values. This model variability needs to be taken into account for the experimental design to guarantee that the Bayesian inverse solution is uniformly informative. In this work we link the numerical stability of the maximum a posterior point and A-optimal experimental design to an observability coefficient that directly describes the influence of the chosen sensors. We propose an algorithm that iteratively chooses the sensor locations to improve this coefficient and thereby decrease the eigenvalues of the posterior covariance matrix. This algorithm exploits the structure of the solution manifold in the hyper-parameter domain via a reduced basis surrogate solution for computational efficiency. We illustrate our results with a steady-state thermal conduction problem.