I believe it may have been a limitation of the API Gateway + Lambda thing. Just checked the docs again and it says there is an "Integration Timeout" as follows:

I believe it may have been a limitation of the API Gateway + Lambda thing. Just checked the docs again and it says there is an "Integration Timeout" as follows: > 30 seconds for all integration types, including Lambda, Lambda proxy, HTTP, HTTP proxy, and AWS integrations. So not Lambda alone, but in conjunction with the API Gateway.