Deep learning models have advanced our ability to predict cell-type-specific chromatin patterns from transcription factor (TF) binding motifs, but their application to perturbed contexts remains limited. We applied transfer learning to predict how concentrations of the dosage-sensitive TFs TWIST1 and SOX9 affect regulatory element (RE) chromatin accessibility in facial progenitor cells, achieving near-experimental accuracy. High-affinity motifs that allow for heterotypic TF co-binding and are concentrated at the center of REs buffer against quantitative changes in TF dosage and predict unperturbed accessibility. Conversely, low-affinity or homotypic binding motifs distributed throughout REs drive sensitive responses with minimal impact on unperturbed accessibility. Both buffering and sensitizing features display purifying selection signatures. We validated these sequence features through reporter assays and demonstrated that TF-nucleosome competition can explain low-affinity motifs' sensitizing effects. This combination of transfer learning and quantitative chromatin response measurements provides a novel approach for uncovering additional layers of the cis-regulatory code.