Large field-of-view (FOV) cameras can simplify and accelerate scene capture because they provide complete coverage with fewer views. However, existing reconstruction pipelines fail to take full advantage of large-FOV input data because they convert input views to perspective images, resulting in stretching that prevents the use of the full image. Additionally, they calibrate lenses using models that do not accurately fit real fisheye lenses in the periphery. We present a new reconstruction pipeline based on Gaussian Splatting that uses a flexible lens model and supports fields of view approaching 180 degrees. We represent lens distortion with a hybrid neural field based on an Invertible ResNet and use a cubemap to render wide-FOV images while retaining the efficiency of the Gaussian Splatting pipeline. Our system jointly optimizes lens distortion, camera intrinsics, camera poses, and scene representations using a loss measured directly against the original input pixels. We present extensive experiments on both synthetic and real-world scenes, demonstrating that our model accurately fits real-world fisheye lenses and that our end-to-end self-calibration approach provides higher-quality reconstructions than existing methods.