Automated fetal brain extraction from full-uterus MRI is a challenging task due to variable head sizes, orientations, complex anatomy, and prevalent artifacts. While deep-learning (DL) models trained on synthetic images have been successful in adult brain extraction, adapting these networks for fetal MRI is difficult due to the sparsity of labeled data, leading to increased false-positive predictions. To address this challenge, we propose a test-time strategy that reduces false positives in networks trained on sparse, synthetic labels. The approach uses a breadth-fine search (BFS) to identify a subvolume likely to contain the fetal brain, followed by a deep-focused sliding window (DFS) search to refine the extraction, pooling predictions to minimize false positives. We train models at different window sizes using synthetic images derived from a small number of fetal brain label maps, augmented with random geometric shapes. Each model is trained on diverse head positions and scales, including cases with partial or no brain tissue. Our framework matches state-of-the-art brain extraction methods on clinical HASTE scans of third-trimester fetuses and exceeds them by up to 5\% in terms of Dice in the second trimester as well as EPI scans across both trimesters. Our results demonstrate the utility of a sliding-window approach and combining predictions from several models trained on synthetic images, for improving brain-extraction accuracy by progressively refining regions of interest and minimizing the risk of missing brain mask slices or misidentifying other tissues as brain.