Do you find that using the JVM in Lambda affects your cold start times? If so, do you have a pre-warm strategy and how does it work? Or maybe it doesn't matter for your application?
Try Micronaut, which is a micro framework specifically designed for superfast startup time and low memory footprint. It also works with GraalVM.
I did when I used Java Lambdas behind API gateway for a largish public API. I ended just setting the HTTP purification tests to run every minute. Solved the issue and let me know how stable it was. Not a huge issue in the end.
Lambda does re-use the JVM instances where it can to alleviate that.
That only matters if your concurrency is one. If you get 10 requests each before the previous one completes you will get cold starts every other call or so.. its manageable, just takes effort.