Might AWS Lambda behind AWS API Gateway (with caching, burst and rate limiting, custom authorizers) behind AWS CloudFront behind AWS WAF and/or CloudFlare work for you?

What is your issue with keeping Lambda instance alive? They mostly stay in standby mode some minutes to keep listening for requests and then it will shutdown whether staying idle - what means next request will boot a Lambda function (container?; It usually takes ~1 second), after that time to receive response is approx. ~100ms.