This is not up to Kagi. Anthropic does not allow you to use your subscription in anything other than what Anthropic created. If you want to use Claude in a third-party app, you have to pay api prices. Any efforts Kagi may make in implimenting this would violate Claude's terms of service -- and they have not been hesitant to ban users who do so.
People who have used OpenClaw with their Claude subscriptions have found themselves suddenly banned for it, and one user was charged api pricing unexpectedly because a file titled HERMES.MD was found in their git commit history. Yes, that was a bug, but Anthropic refused to refund it until after it went public and people started talking about it.