Context. A significant challenge in the study of transient astrophysical phenomena is the identification of bogus events, among which human-made Earth-orbiting satellites and debris remain major contaminants. Existing pipelines can effectively identify satellite trails, but they often miss more complex signatures, such as collections of satellite glints. In the Rubin Observatory era, the scale of operations will increase tenfold with respect to its precursor, the Zwicky Transient Facility (ZTF), requiring crucial improvements in classification purity, data compression for informative alerts, and pipeline speed. Aims. We explore the use of a 2D Fast Fourier Transform (FFT) on difference images as a tool to improve satellite-detection machine learning algorithms. Methods. Using the Automatic Learning for the Rapid Classification of Events (ALeRCE) single-stamp classifier as a baseline, we adapted its architecture to receive a cutout of the FFT of the difference image, in addition to the three (science, reference, difference) ZTF image cutouts (hereafter stamps). We explored various stamp sizes and resolutions, assessing the benefits of incorporating FFT images, particularly when data compression is critical due to alert size limitations and pipeline speed constraints (e.g., in large-scale surveys such as the Legacy Survey of Space and Time). Results. The inclusion of the FFT can significantly improve satellite detection performance. The most notable improvement occurred in the smallest field-of-view model (16 ''), whose satellite classification accuracy increased from (72.0 +/- 2.9)% to (87.8 +/- 1.3)% after including the FFT, computed from the full 63 '' difference images. This demonstrates the effectiveness of FFT in compressing and extracting relevant large-scale satellite features. However, the FFT alone did not fully match the accuracy achieved by the full 63 '', (95.9 +/- 1.3)% and multiscale (90.6 +/- 0.8)% models, highlighting the complementary importance of contextual spatial information. Conclusions. We show how FFTs can be leveraged to cull satellite and space debris signatures from alert streams.