Docker Service in Swarm cannot connect to AWS RDS when a port is published

The company I'm working for recently decided to deploy a new application with docker swarm on AWS using EC2 instances. We set up a cluster of three EC2 instances as nodes (one manager, two workers) and we use stack to deploy the services.
The problem is that one of the services, a django app, runs into a timeout when trying to connect to the postgres database that runs on RDS in the same VPC. But ONLY when the service publishes a port.
A service that doesn't publish any port can connect to the DB just fine.

The RDS endpoint gets resolved to the proper IP, so it shouldn't be a DNS issue and the containers can connect to the internet. The services are also able to talk to each other on the different nodes.
There also shouldn't be a problem with the security group definition of the db, because the EC2 instances themselves can get a connection to the DB.

Further, the services can connect to things that are running on other instances within the VPC.
It seems that it has something to do with swarm (and overlay networks) as running the app inside a normal container with a bridge network doesn't cause any problems.
Stack doesn't seem to be the problem, because even when creating the services manually, the issue still persists.

We are using Docker CE version 19.03.8, on Ubuntu 18.04.4 LTS and docker-compose version 3.