This paper examines the challenge of shaping a battery's input trajectory to (i) maximize its Fisher parameter identifiability while (ii) achieving robustness to parameter uncertainties. The paper is motivated by earlier research showing that the speed and accuracy with which battery parameters can be estimated both improve significantly when battery inputs are optimized for Fisher identifiability. Previous research performs this trajectory optimization for a known nominal parameter set. This creates a tautology where accurate parameter identification is a prerequisite for Fisher identifiability optimization. In contrast, this paper presents an iterative scheme that: (i) uses prior parameter probability distributions to create a weighted Fisher metric; (ii) optimizes the battery input trajectory for this metric using a genetic algorithm; (iii) applies the resulting input trajectory to the battery; (iv) estimates battery parameters using a Bayesian particle filter; (v) re-computes the weighted Fisher information metric using the resulting posterior parameter distribution; and (vi) repeats this process until convergence. This approach builds on well-established ideas from the estimation literature, and applies them to the battery domain for the first time. Simulation studies highlight the ability of this iterative algorithm to converge quickly towards the correct battery parameter values, despite large initial parameter uncertainties.