Rate exceeded error for Lambda in Step Function

I'm pretty new to this architecture and it is SQS->Lambda (just intermediary) ->Step Function (comprises Lambdas). This error comes up if I drop 1k messages into SQS quickly. When I first encountered this, I tried to manage the rate of Step Function invocations by limiting the Lambda's reserved concurrency to 10 while the Step Function has unreserved concurrency 200. Then, the error still happens if the Step Function Lambdas are cold, but ok if they're warm. What are the solutions to this and what $ cost tradeoff do I need to consider?