I haven't tried with a NAT Gateway.
This is what I know.
If the response is small, the request duration is small.
Cloudfront > Gateway > Lambda > RDS (PostgreSQL)
Response:
1k | 20ms-60ms
10k | 50ms-90ms
100k | 150ms-200ms
350k | 200ms-450ms
That's a rough gauge of what I've experienced.
I think the throughput on the gateway is the bottleneck.