Frequency-invariant (FI) beamformers are widely used in audio and acoustic applications. A key challenge is achieving frequency and rotational invariance with concentric circular arrays under hardware constraints. This paper proposes a neural-network-based end-to-end framework for sparse FI beamformer design that jointly optimizes array geometry and per-ring sensor assignment. Simulation results show that the proposed model achieves competitive low-frequency robustness and improved high-frequency robustness compared to state-of-the-art designs, while maintaining a frequency-invariant beampattern with low rotational error and reasonable computational and hardware resources. The framework also supports flexible control of the trade-off between Directivity Factor (DF) and White Noise Gain (WNG).