I’ve now gotten an answer from internal support. Regardless of total connection cap, Render has a soft cap of 5 connections per second from external sources (eg, a server on AWS), likely causing my issue. Posting this here in case others run into the same issue.
For future folks who run into this issue, my solution is to bite the bullet and move Redis over to a major cloud provider — I’ve implemented connection pooling, etc, but at any reasonable background job concurrency, I’m going to smash that 5 connections / second in bursts.