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They certainly have an enterprise plan?

Databricks is most certainly getting charged API pricing no matter what harness they are using. OpenAI and Anthropic models are so sought after right now that they set the terms even at the world's biggest companies, there is not a chance to get a special agreement for subscription pricing.

You can use both of those harnesses without going through subscription. That is a native feature in both Codex & Claude Code, even for non-enterprise customers.