Alzheimer's disease (AD) diagnosis requires sequential evidence acquisition under heterogeneous test costs and patient burden. Fixed-modality predictors do not jointly decide which test to acquire or when the available evidence is sufficient for diagnosis. We propose SCOPE-AD (Sequential Cost-Aware Ordinal-Belief Planning with Energy-Based Models for Diagnostic Agents) for cost-aware classification of cognitively normal (CN), mild cognitive impairment (MCI), and AD cases. A mask-aware ordinal model represents uncertainty along the ordered CN--MCI--AD continuum. Retrospective training records provide sampled Bellman targets for an energy-based teacher, whose action distributions are distilled into a Qwen policy. At deployment, the agent selects acquisition or diagnosis actions under availability and budget constraints without access to unacquired values. After each acquisition, the evidence and ordinal belief are updated before the next decision. On ADNI, SCOPE-AD achieves 77.70\% Macro-F1 at an average acquisition cost of \$50.46, exceeding the strongest evaluated baseline by 9.34 percentage points. Full-modality evaluation raises Macro-F1 by only 1.89 points while increasing acquisition cost by 116.7 times. These results support selective acquisition for cost-effective diagnosis.

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