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You can't do that directly; but what you could do is use a custom authorizer which stores the number of calls from each client in a persistent manner (DynamoDB perhaps?) and then allows or disallows access based on the number of requests.
It's extra work to do; the alternative is to use the REST API Gateway service.

This will only work if you disable authorization caching, which will increase the cost as API GW will invoke the Lambda function for each request.