Tracing API Gateway 5xx errors

We have an API Gateway REST API (very standard, no auth, nothing interesting) which calls a bunch of Lambdas

Since yesterday ~12:00 GMT, we've been receiving a large number of 5xx errors, triggering one of our slack notifications. None of our lambdas have any errors (searched via metric filter and looking at the Cloudwatch metrics for lambdas), and the API GW execution logs are also clean. As far as we can see, all of our services are functioning normally, our monitoring services are all green, and we've had no client reports of outages/degraded services. There have been no deployments since last week, and there's no change in any other metric across the system. API GW only tells us that this is 5xx and not something more specific (such as a timeout).

How can we debug this? Ideally we'd like to see what the exact status/message is from these errors and where they are coming from. It could easily be that someone is hitting our API with large payloads or bad data, but we can't tell as it's not even getting to our lambdas.