The automated detection of Alzheimer's Disease (AD) is increasingly critical for global healthcare systems, as conventional diagnostic pathways remain subjective, resource-intensive, and poorly scalable to resource-constrained clinical settings such as those prevalent in Indonesia. This study presents a rigorous comparative evaluation of three deep learning architectures - a custom Convolutional Neural Network (CNN), ResNet-50, and Vision Transformer (ViT/B-16)- for binary AD classification from Axial T1-weighted MRI. To assess genuine generalizability rather than in-distribution performance, a cross-dataset validation strategy was adopted: models were trained on the public dataset, i.e., the Falah/HuggingFace dataset, and then evaluated on the private dataset collected from Rumah Sakit Hasan Sadikin (RSHS), Bandung, Indonesia. To eliminate hyperparameter bias as a confounding variable, a systematic Grid Search across 27 configurations was conducted, combining three optimizers (Adam, SGD, and AdamW) with three learning rates (0.001, 0.0001, and 0.00001) per architecture. ResNet-50 with SGD at lr = 0.0001 emerged as the optimal configuration, achieving the highest internal accuracy of 0.989 (F1 = 0.980) and retaining a robust external accuracy of 0.823 (F1 = 0.825), surpassing all six prior benchmarked studies, none of which conducted external clinical validation. ViT exhibited severe domain sensitivity, collapsing from 0.985 internal to 0.617 external accuracy, a 37.36% degradation attributable to attention miscalibration under distribution shift. The Custom CNN achieved a moderate external accuracy of 0.653, outperforming ViT by 5.83% and confirming the structural advantage of convolutional inductive biases in limited-data regimes. These findings indicate that CNN-based architectures offer a safer, computationally more efficient, and clinically applicable solution for AD diagnosis compared to Transformer models, which rely heavily on large amounts of data.