Best one I found is Normalized Mutual Information at 95%. It's a little more complex, but you could still compute it relatively quickly on binarized images.

    NMI skimage: ~30 seconds, 95% accuracy
    NMI numba *: ~0.6 seconds, 95% accuracy

Here's the code, courtesy of ChatGPT:

    @njit(cache=True)
    def compute_joint(img1, img2):
        stacked = 2 * img1.ravel() + img2.ravel()
        return np.bincount(stacked, minlength=4).reshape(2, 2)

    @njit(cache=True)
    def entropy(counts, total):
        # Convert counts to probabilities
        p = counts / total
        # Only calculate for non-zero entries to avoid log(0)
        return -np.sum(p[p > 0] * np.log2(p[p > 0]))

    @njit(cache=True)
    def normalized_mutual_information(img1, img2):
        joint_counts = compute_joint(img1, img2)
        total_pixels = 28*28
        
        # Marginal counts
        c1 = np.sum(joint_counts, axis=1)
        c2 = np.sum(joint_counts, axis=0)
        
        # Compute entropies directly from counts
        h1 = entropy(c1, total_pixels)
        h2 = entropy(c2, total_pixels)
        joint_entropy = entropy(joint_counts.flatten(), total_pixels)
        
        mutual_info = h1 + h2 - joint_entropy
        return 2 * mutual_info / (h1 + h2)