I got one of our DB experts interested, and he helped me add some additional postgres logging params (as Weedpacket referred to above), and we'll start diving into the results tomorrow (or whenever) to see if we can narrow down where the actual lag is. He said he'd set up pgbouncer a couple times, so if we decide that's a good option, I'll have a good head start.
We use Docker containers for our application, so I suspect either it would be installed on each web container, or else as its own container on AWS/ECS.