Going to EKS would take less time than exploring hacks.
Making a switch to EKS doesn't mean that you are stuck with EKS forever, and can set you up nicely to make the transition to a more complex/customized cluster implementation/deployment (likely still on AWS, think kops) easier if you find it's necessary for your use case in the future.
There seem to be some other intermittent network issues as well. I just moved some stuff for a client off EKS and back onto ECS because it wasn't worth troubleshooting.
That said, I am not a fan of ECS at all. The scheduler is slow and it does really just take way too long for it to even start acting on new deployments.