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The Fréchet Inception Distance (FID) is a standard metric for evaluating generative image models, but its underlying model Inception-v3 is trained on natural images and is poorly suited to handwritten text. In this work, we investigate how replacing the feature extractor with models more relevant to the handwriting domain affects the resulting Fréchet distance curve across different levels of image degradation. Three types of models are evaluated: TrOCR and AttentionHTR (domain-specific) and DINOv3 (general-purpose), each combined with mean, max, or CLS pooling where applicable. The candidates are tested on a synthetic dataset derived from the Saint Gall manuscripts with six controlled degradation types: blur, dilation, erosion, noise, overtext, and rotation. Using monotonicity and human judgment as evaluation criteria, we find that the Imgur5K variant of AttentionHTR with mean pooling produces the most consistent response across all degradation types, with the SLL variant of TrOCR using CLS pooling as a strong alternative. The proposed metric, called Handwriting Fréchet Distance, is compared to FID, HWD, SSIM, and PSNR, and is shown to better reflect the perceived severity of degradations in handwritten text. The metric is available at https://github.com/HFD-uu/HFD.